/[escript]/trunk/escript/py_src/flows.py
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revision 2549 by jfenwick, Mon Jul 20 06:43:47 2009 UTC revision 3502 by gross, Thu Apr 28 05:06:24 2011 UTC
# Line 1  Line 1 
1    # -*- coding: utf-8 -*-
2  ########################################################  ########################################################
3  #  #
4  # Copyright (c) 2003-2009 by University of Queensland  # Copyright (c) 2003-2010 by University of Queensland
5  # Earth Systems Science Computational Center (ESSCC)  # Earth Systems Science Computational Center (ESSCC)
6  # http://www.uq.edu.au/esscc  # http://www.uq.edu.au/esscc
7  #  #
# Line 10  Line 11 
11  #  #
12  ########################################################  ########################################################
13    
14  __copyright__="""Copyright (c) 2003-2009 by University of Queensland  __copyright__="""Copyright (c) 2003-2010 by University of Queensland
15  Earth Systems Science Computational Center (ESSCC)  Earth Systems Science Computational Center (ESSCC)
16  http://www.uq.edu.au/esscc  http://www.uq.edu.au/esscc
17  Primary Business: Queensland, Australia"""  Primary Business: Queensland, Australia"""
# Line 21  __url__="https://launchpad.net/escript-f Line 22  __url__="https://launchpad.net/escript-f
22  """  """
23  Some models for flow  Some models for flow
24    
25  @var __author__: name of author  :var __author__: name of author
26  @var __copyright__: copyrights  :var __copyright__: copyrights
27  @var __license__: licence agreement  :var __license__: licence agreement
28  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
29  @var __version__: version  :var __version__: version
30  @var __date__: date of the version  :var __date__: date of the version
31  """  """
32    
33  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
34    
35  from escript import *  import escript
36  import util  import util
37  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
38  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
39    
40  class DarcyFlow(object):  class DarcyFlow(object):
41      """     """
42      solves the problem     solves the problem
43      
44      M{u_i+k_{ij}*p_{,j} = g_i}     *u_i+k_{ij}*p_{,j} = g_i*
45      M{u_{i,i} = f}     *u_{i,i} = f*
46      
47      where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability,     where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
48      
49      @note: The problem is solved in a least squares formulation.     :cvar SIMPLE: simple solver
50      """     :cvar POST: solver using global postprocessing of flux
51       :cvar STAB: solver uses (non-symmetric) stabilization
52      def __init__(self, domain, weight=None, useReduced=False, adaptSubTolerance=True):     :cvar SYMSTAB: solver uses symmetric stabilization
53          """     """
54          initializes the Darcy flux problem     SIMPLE="SIMPLE"
55          @param domain: domain of the problem     POST="POST"
56          @type domain: L{Domain}     STAB="STAB"
57      @param useReduced: uses reduced oreder on flux and pressure     SYMSTAB="SYMSTAB"
58      @type useReduced: C{bool}     def __init__(self, domain, useReduced=False, solver="SYMSTAB", verbose=False, w=1.):
59      @param adaptSubTolerance: switches on automatic subtolerance selection        """
60      @type adaptSubTolerance: C{bool}            initializes the Darcy flux problem
61          """        :param domain: domain of the problem
62          self.domain=domain        :type domain: `Domain`
63          if weight == None:        :param useReduced: uses reduced oreder on flux and pressure
64             s=self.domain.getSize()        :type useReduced: ``bool``
65             self.__l=(3.*util.longestEdge(self.domain)*s/util.sup(s))**2        :param solver: solver method
66             # self.__l=(3.*util.longestEdge(self.domain))**2        :type solver: in [`DarcyFlow.SIMPLE`, `DarcyFlow.POST', `DarcyFlow.STAB`, `DarcyFlow.SYMSTAB` ]
67             # self.__l=(0.1*util.longestEdge(self.domain)*s/util.sup(s))**2        :param verbose: if ``True`` some information on the iteration progress are printed.
68          else:        :type verbose: ``bool``
69             self.__l=weight        :param w: weighting factor for `DarcyFlow.POST` solver
70          self.__pde_v=LinearPDESystem(domain)        :type w: ``float``
71          if useReduced: self.__pde_v.setReducedOrderOn()        
72          self.__pde_v.setSymmetryOn()        """
73          self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain)))        self.domain=domain
74          self.__pde_p=LinearSinglePDE(domain)        self.solver=solver
75          self.__pde_p.setSymmetryOn()        self.useReduced=useReduced
76          if useReduced: self.__pde_p.setReducedOrderOn()        self.verbose=verbose
77          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))        self.scale=1.
78          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))        
79          self.setTolerance()        
80          self.setAbsoluteTolerance()        self.__pde_v=LinearPDESystem(domain)
81      self.__adaptSubTolerance=adaptSubTolerance        self.__pde_v.setSymmetryOn()
82      self.verbose=False        if self.useReduced: self.__pde_v.setReducedOrderOn()
83      def getSolverOptionsFlux(self):        self.__pde_p=LinearSinglePDE(domain)
84      """        self.__pde_p.setSymmetryOn()
85      Returns the solver options used to solve the flux problems        if self.useReduced: self.__pde_p.setReducedOrderOn()
86              
87      M{(I+D^*D)u=F}        if self.solver  == self.SIMPLE:
88             if self.verbose: print "DarcyFlow: simple solver is used."
89      @return: L{SolverOptions}           self.__pde_v.setValue(D=util.kronecker(self.domain.getDim()))
90      """        elif self.solver  == self.POST:
91      return self.__pde_v.getSolverOptions()       self.w=w
92      def setSolverOptionsFlux(self, options=None):       if util.inf(w)<0.:
93      """          raise ValueError,"Weighting factor must be non-negative."
94      Sets the solver options used to solve the flux problems       if self.verbose: print "DarcyFlow: global postprocessing of flux is used."
95              elif self.solver  == self.STAB:
96      M{(I+D^*D)u=F}        if self.verbose: print "DarcyFlow: (non-symmetric) stabilization is used."
97              elif  self.solver  == self.SYMSTAB:
98      If C{options} is not present, the options are reset to default        if self.verbose: print "DarcyFlow: symmetric stabilization is used."
99      @param options: L{SolverOptions}        else:
100      @note: if the adaption of subtolerance is choosen, the tolerance set by C{options} will be overwritten before the solver is called.      raise ValueError,"unknown solver %s"%self.solver
101      """        self.__f=escript.Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
102      return self.__pde_v.setSolverOptions(options)        self.__g=escript.Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
103      def getSolverOptionsPressure(self):        self.location_of_fixed_pressure = escript.Scalar(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
104      """        self.location_of_fixed_flux = escript.Vector(0, self.__pde_v.getFunctionSpaceForCoefficient("q"))
105      Returns the solver options used to solve the pressure problems        self.setTolerance()
106            
107      M{(Q^*Q)p=Q^*G}          
108       def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
109          """
110          assigns values to model parameters
111    
112          :param f: volumetic sources/sinks
113          :type f: scalar value on the domain (e.g. `escript.Data`)
114          :param g: flux sources/sinks
115          :type g: vector values on the domain (e.g. `escript.Data`)
116          :param location_of_fixed_pressure: mask for locations where pressure is fixed
117          :type location_of_fixed_pressure: scalar value on the domain (e.g. `escript.Data`)
118          :param location_of_fixed_flux:  mask for locations where flux is fixed.
119          :type location_of_fixed_flux: vector values on the domain (e.g. `escript.Data`)
120          :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
121          :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
122    
123          :note: the values of parameters which are not set by calling ``setValue`` are not altered.
124          :note: at any point on the boundary of the domain the pressure
125                 (``location_of_fixed_pressure`` >0) or the normal component of the
126                 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
127                 is along the *x_i* axis.
128    
129          """
130          if location_of_fixed_pressure!=None:
131               self.location_of_fixed_pressure=util.wherePositive(location_of_fixed_pressure)
132               self.__pde_p.setValue(q=self.location_of_fixed_pressure)
133          if location_of_fixed_flux!=None:
134              self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
135              self.__pde_v.setValue(q=self.location_of_fixed_flux)
136          
137                
138          # pressure  is rescaled by the factor 1/self.scale
139          if permeability!=None:
140            
141      @return: L{SolverOptions}       perm=util.interpolate(permeability,self.__pde_v.getFunctionSpaceForCoefficient("A"))
142      """           V=util.vol(self.domain)
143      return self.__pde_p.getSolverOptions()           l=V**(1./self.domain.getDim())
144      def setSolverOptionsPressure(self, options=None):          
145      """       if perm.getRank()==0:
146      Sets the solver options used to solve the pressure problems          perm_inv=(1./perm)
147                    self.scale=util.integrate(perm_inv)/V*l
148      M{(Q^*Q)p=Q^*G}          perm_inv=perm_inv*((1./self.scale)*util.kronecker(self.domain.getDim()))
149                perm=perm*(self.scale*util.kronecker(self.domain.getDim()))
150      If C{options} is not present, the options are reset to default          
151      @param options: L{SolverOptions}          
152      @note: if the adaption of subtolerance is choosen, the tolerance set by C{options} will be overwritten before the solver is called.       elif perm.getRank()==2:
153      """          perm_inv=util.inverse(perm)
154      return self.__pde_p.setSolverOptions(options)              self.scale=util.sqrt(util.integrate(util.length(perm_inv)**2)/V)*l
155            perm_inv*=(1./self.scale)
156      def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):          perm=perm*self.scale
157          """       else:
158          assigns values to model parameters          raise ValueError,"illegal rank of permeability."
159            
160          @param f: volumetic sources/sinks       self.__permeability=perm
161          @type f: scalar value on the domain (e.g. L{Data})       self.__permeability_inv=perm_inv
162          @param g: flux sources/sinks       if self.verbose: print "DarcyFlow: scaling factor for pressure is %e."%self.scale
         @type g: vector values on the domain (e.g. L{Data})  
         @param location_of_fixed_pressure: mask for locations where pressure is fixed  
         @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})  
         @param location_of_fixed_flux:  mask for locations where flux is fixed.  
         @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})  
         @param permeability: permeability tensor. If scalar C{s} is given the tensor with  
                              C{s} on the main diagonal is used. If vector C{v} is given the tensor with  
                              C{v} on the main diagonal is used.  
         @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})  
   
         @note: the values of parameters which are not set by calling C{setValue} are not altered.  
         @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)  
                or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal  
                is along the M{x_i} axis.  
         """  
         if f !=None:  
            f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))  
            if f.isEmpty():  
                f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))  
            else:  
                if f.getRank()>0: raise ValueError,"illegal rank of f."  
            self.__f=f  
         if g !=None:  
            g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))  
            if g.isEmpty():  
              g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))  
            else:  
              if not g.getShape()==(self.domain.getDim(),):  
                raise ValueError,"illegal shape of g"  
            self.__g=g  
   
         if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)  
         if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)  
   
         if permeability!=None:  
            perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))  
            if perm.getRank()==0:  
                perm=perm*util.kronecker(self.domain.getDim())  
            elif perm.getRank()==1:  
                perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm  
                for i in range(self.domain.getDim()): perm[i,i]=perm2[i]  
            elif perm.getRank()==2:  
               pass  
            else:  
               raise ValueError,"illegal rank of permeability."  
            self.__permeability=perm  
            self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))  
   
     def setTolerance(self,rtol=1e-4):  
         """  
         sets the relative tolerance C{rtol} used to terminate the solution process. The iteration is terminated if  
   
         M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }  
   
         where C{atol} is an absolut tolerance (see L{setAbsoluteTolerance}), M{|f|^2 = integrate(length(f)^2)} and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.  
   
         @param rtol: relative tolerance for the pressure  
         @type rtol: non-negative C{float}  
         """  
         if rtol<0:  
             raise ValueError,"Relative tolerance needs to be non-negative."  
         self.__rtol=rtol  
     def getTolerance(self):  
         """  
         returns the relative tolerance  
   
         @return: current relative tolerance  
         @rtype: C{float}  
         """  
         return self.__rtol  
   
     def setAbsoluteTolerance(self,atol=0.):  
         """  
         sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if  
   
         M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }  
   
         where C{rtol} is an absolut tolerance (see L{setTolerance}), M{|f|^2 = integrate(length(f)^2)} and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.  
   
         @param atol: absolute tolerance for the pressure  
         @type atol: non-negative C{float}  
         """  
         if atol<0:  
             raise ValueError,"Absolute tolerance needs to be non-negative."  
         self.__atol=atol  
     def getAbsoluteTolerance(self):  
        """  
        returns the absolute tolerance  
         
        @return: current absolute tolerance  
        @rtype: C{float}  
        """  
        return self.__atol  
     def getSubProblemTolerance(self):  
     """  
     Returns a suitable subtolerance  
     @type: C{float}  
     """  
     return max(util.EPSILON**(0.75),self.getTolerance()**2)  
     def setSubProblemTolerance(self):  
          """  
          Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected.  
          """  
      if self.__adaptSubTolerance:  
          sub_tol=self.getSubProblemTolerance()  
              self.getSolverOptionsFlux().setTolerance(sub_tol)  
          self.getSolverOptionsFlux().setAbsoluteTolerance(0.)  
          self.getSolverOptionsPressure().setTolerance(sub_tol)  
          self.getSolverOptionsPressure().setAbsoluteTolerance(0.)  
          if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol  
   
     def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10):  
          """  
          solves the problem.  
   
          The iteration is terminated if the residual norm is less then self.getTolerance().  
   
          @param u0: initial guess for the flux. At locations in the domain marked by C{location_of_fixed_flux} the value of C{u0} is kept unchanged.  
          @type u0: vector value on the domain (e.g. L{Data}).  
          @param p0: initial guess for the pressure. At locations in the domain marked by C{location_of_fixed_pressure} the value of C{p0} is kept unchanged.  
          @type p0: scalar value on the domain (e.g. L{Data}).  
          @param verbose: if set some information on iteration progress are printed  
          @type verbose: C{bool}  
          @return: flux and pressure  
          @rtype: C{tuple} of L{Data}.  
   
          @note: The problem is solved as a least squares form  
   
          M{(I+D^*D)u+Qp=D^*f+g}  
          M{Q^*u+Q^*Qp=Q^*g}  
   
          where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.  
          We eliminate the flux form the problem by setting  
   
          M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux  
   
          form the first equation. Inserted into the second equation we get  
   
          M{Q^*(I-(I+D^*D)^{-1})Qp= Q^*(g-(I+D^*D)^{-1}(D^*f+g))} with p=p0  on location_of_fixed_pressure  
   
          which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step  
          PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.  
          """  
          self.verbose=verbose  
          rtol=self.getTolerance()  
          atol=self.getAbsoluteTolerance()  
      self.setSubProblemTolerance()  
163            
164           num_corrections=0       if self.solver  == self.SIMPLE:
165           converged=False          self.__pde_p.setValue(A=self.__permeability)
166           p=p0       elif self.solver  == self.POST:
167           norm_r=None          self.__pde_p.setValue(A=self.__permeability)
168           while not converged:          k=util.kronecker(self.domain.getDim())
169                 v=self.getFlux(p, fixed_flux=u0)          self.lamb = self.w*util.length(perm_inv)*l
170                 Qp=self.__Q(p)          self.__pde_v.setValue(D=self.__permeability_inv, A=self.lamb*self.domain.getSize()*util.outer(k,k))
171                 norm_v=self.__L2(v)       elif self.solver  == self.STAB:
172                 norm_Qp=self.__L2(Qp)          self.__pde_p.setValue(A=0.5*self.__permeability)
173                 if norm_v == 0.:          self.__pde_v.setValue(D=0.5*self.__permeability_inv)
174                    if norm_Qp == 0.:       elif  self.solver  == self.SYMSTAB:
175                       return v,p          self.__pde_p.setValue(A=0.5*self.__permeability)
176                    else:          self.__pde_v.setValue(D=0.5*self.__permeability_inv)
177                      fac=norm_Qp  
178                 else:        if g != None:
179                    if norm_Qp == 0.:      g=util.interpolate(g, self.__pde_v.getFunctionSpaceForCoefficient("Y"))
180                      fac=norm_v      if g.isEmpty():
181                    else:            g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
182                      fac=2./(1./norm_v+1./norm_Qp)      else:
183                 ATOL=(atol+rtol*fac)          if not g.getShape()==(self.domain.getDim(),): raise ValueError,"illegal shape of g"
184                 if self.verbose:      self.__g=g
185                      print "DarcyFlux: L2 norm of v = %e."%norm_v        if f !=None:
186                      print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp       f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
187                      print "DarcyFlux: L2 defect u = %e."%(util.integrate(util.length(self.__g-util.interpolate(v,Function(self.domain))-Qp)**2)**(0.5),)       if f.isEmpty():      
188                      print "DarcyFlux: L2 defect div(v) = %e."%(util.integrate((self.__f-util.div(v))**2)**(0.5),)            f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
189                      print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL       else:
190                 if norm_r == None or norm_r>ATOL:           if f.getRank()>0: raise ValueError,"illegal rank of f."
191                     if num_corrections>max_num_corrections:       self.__f=f
192                           raise ValueError,"maximum number of correction steps reached."     def getSolverOptionsFlux(self):
193                     p,r, norm_r=PCG(self.__g-util.interpolate(v,Function(self.domain))-Qp,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.5*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)        """
194                     num_corrections+=1        Returns the solver options used to solve the flux problems
195                 else:        :return: `SolverOptions`
196                     converged=True        """
197           return v,p        return self.__pde_v.getSolverOptions()
198      def __L2(self,v):        
199           return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))     def setSolverOptionsFlux(self, options=None):
200          """
201      def __Q(self,p):        Sets the solver options used to solve the flux problems
202            return util.tensor_mult(self.__permeability,util.grad(p))        If ``options`` is not present, the options are reset to default
203          :param options: `SolverOptions`
204      def __Aprod(self,dp):        """
205            if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator"        return self.__pde_v.setSolverOptions(options)
206            Qdp=self.__Q(dp)      
207            self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())     def getSolverOptionsPressure(self):
208            du=self.__pde_v.getSolution()        """
209            # self.__pde_v.getOperator().saveMM("proj.mm")        Returns the solver options used to solve the pressure problems
210            return Qdp+du        :return: `SolverOptions`
211      def __inner_GMRES(self,r,s):        """
212           return util.integrate(util.inner(r,s))        return self.__pde_p.getSolverOptions()
213          
214      def __inner_PCG(self,p,r):     def setSolverOptionsPressure(self, options=None):
215           return util.integrate(util.inner(self.__Q(p), r))        """
216          Sets the solver options used to solve the pressure problems
217      def __Msolve_PCG(self,r):        If ``options`` is not present, the options are reset to default
218        if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner"        
219            self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data())        :param options: `SolverOptions`
220            # self.__pde_p.getOperator().saveMM("prec.mm")        :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
221            return self.__pde_p.getSolution()        """
222          return self.__pde_p.setSolverOptions(options)
223      def getFlux(self,p=None, fixed_flux=Data()):        
224          """     def setTolerance(self,rtol=1e-4):
225          returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}        """
226          on locations where C{location_of_fixed_flux} is positive (see L{setValue}).        sets the relative tolerance ``rtol`` for the pressure for the stabelized solvers.
227          Note that C{g} and C{f} are used, see L{setValue}.        
228          :param rtol: relative tolerance for the pressure
229          @param p: pressure.        :type rtol: non-negative ``float``
230          @type p: scalar value on the domain (e.g. L{Data}).        """
231          @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.        if rtol<0:
232          @type fixed_flux: vector values on the domain (e.g. L{Data}).       raise ValueError,"Relative tolerance needs to be non-negative."
233          @param tol: relative tolerance to be used.        self.__rtol=rtol
234          @type tol: positive C{float}.        
235          @return: flux     def getTolerance(self):
236          @rtype: L{Data}        """
237          @note: the method uses the least squares solution M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}}        returns the relative tolerance
238                 for the permeability M{k_{ij}}        :return: current relative tolerance
239          """        :rtype: ``float``
240      self.setSubProblemTolerance()        """
241          g=self.__g        return self.__rtol
242          f=self.__f        
243          self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)     def solve(self,u0,p0, max_iter=100, iter_restart=20):
244          if p == None:        """
245             self.__pde_v.setValue(Y=g)        solves the problem.
246          else:        
247             self.__pde_v.setValue(Y=g-self.__Q(p))        The iteration is terminated if the residual norm is less then self.getTolerance().
248          return self.__pde_v.getSolution()  
249          :param u0: initial guess for the flux. At locations in the domain marked by ``location_of_fixed_flux`` the value of ``u0`` is kept unchanged.
250          :type u0: vector value on the domain (e.g. `escript.Data`).
251          :param p0: initial guess for the pressure. At locations in the domain marked by ``location_of_fixed_pressure`` the value of ``p0`` is kept unchanged.
252          :type p0: scalar value on the domain (e.g. `escript.Data`).
253          :param max_iter: maximum number of (outer) iteration steps for the stabilization solvers,
254          :type max_iter: ``int``
255          :param iter_restart: number of steps after which the iteration is restarted. The larger ``iter_restart`` the larger the required memory.
256                               A small value for ``iter_restart`` may require a large number of iteration steps or may even lead to a failure
257                               of the iteration. ``iter_restart`` is relevant for the stabilization solvers only.
258          :type iter_restart: ``int``
259          :return: flux and pressure
260          :rtype: ``tuple`` of `escript.Data`.
261    
262          """
263          # rescale initial guess:
264          p0=p0/self.scale
265          if self.solver  == self.SIMPLE or self.solver  == self.POST :
266            self.__pde_p.setValue(X=self.__g ,
267                                  Y=self.__f,
268                                  y=-util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux),
269                                  r=p0)
270            p=self.__pde_p.getSolution()
271            u = self.getFlux(p, u0)
272          elif  self.solver  == self.STAB:
273        u,p = self.__solve_STAB(u0,p0, max_iter, iter_restart)
274          elif  self.solver  == self.SYMSTAB:
275        u,p = self.__solve_SYMSTAB(u0,p0, max_iter, iter_restart)
276        
277          if self.verbose:
278            KGp=util.tensor_mult(self.__permeability,util.grad(p))
279            def_p=self.__g-(u+KGp)
280            def_v=self.__f-util.div(u, self.__pde_v.getFunctionSpaceForCoefficient("X"))
281            print "DarcyFlux: |g-u-K*grad(p)|_2 = %e (|u|_2 = %e)."%(self.__L2(def_p),self.__L2(u))
282            print "DarcyFlux: |f-div(u)|_2 = %e (|grad(u)|_2 = %e)."%(self.__L2(def_v),self.__L2(util.grad(u)))
283          #rescale result
284          p=p*self.scale
285          return u,p
286          
287       def getFlux(self,p, u0=None):
288            """
289            returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
290            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
291            Notice that ``g`` and ``f`` are used, see `setValue`.
292    
293            :param p: pressure.
294            :type p: scalar value on the domain (e.g. `escript.Data`).
295            :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
296            :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
297            :return: flux
298            :rtype: `escript.Data`
299            """
300            if self.solver  == self.SIMPLE or self.solver  == self.POST  :
301                KGp=util.tensor_mult(self.__permeability,util.grad(p))
302                self.__pde_v.setValue(Y=self.__g-KGp, X=escript.Data())
303                if u0 == None:
304               self.__pde_v.setValue(r=escript.Data())
305            else:
306               self.__pde_v.setValue(r=u0)
307                u= self.__pde_v.getSolution()
308        elif self.solver  == self.POST:
309                self.__pde_v.setValue(Y=util.tensor_mult(self.__permeability_inv,self.__g)-util.grad(p),
310                                      X=self.lamb * self.__f * util.kronecker(self.domain.getDim()))
311                if u0 == None:
312               self.__pde_v.setValue(r=escript.Data())
313            else:
314               self.__pde_v.setValue(r=u0)
315                u= self.__pde_v.getSolution()
316        elif self.solver  == self.STAB:
317             gp=util.grad(p)
318             self.__pde_v.setValue(Y=0.5*(util.tensor_mult(self.__permeability_inv,self.__g)+gp),
319                                   X= p * util.kronecker(self.domain.getDim()),
320                                   y= - p * self.domain.getNormal())                          
321             if u0 == None:
322               self.__pde_v.setValue(r=escript.Data())
323             else:
324               self.__pde_v.setValue(r=u0)
325             u= self.__pde_v.getSolution()
326        elif  self.solver  == self.SYMSTAB:
327             gp=util.grad(p)
328             self.__pde_v.setValue(Y=0.5*(util.tensor_mult(self.__permeability_inv,self.__g)-gp),
329                                   X= escript.Data() ,
330                                   y= escript.Data() )                          
331             if u0 == None:
332               self.__pde_v.setValue(r=escript.Data())
333             else:
334               self.__pde_v.setValue(r=u0)
335             u= self.__pde_v.getSolution()
336        return u
337          
338        
339       def __solve_STAB(self, u0, p0, max_iter, iter_restart):
340              # p0 is used as an initial guess
341          u=self.getFlux(p0, u0)  
342              self.__pde_p.setValue( Y=self.__f-util.div(u),
343                                     X=0.5*(self.__g - u - util.tensor_mult(self.__permeability,util.grad(p0)) ),
344                                     y= escript.Data(),
345                                     r=escript.Data())
346          dp=self.__pde_p.getSolution()
347          p=GMRES(dp,
348                  self.__STAB_Aprod,
349              p0,
350              self.__inner,
351              atol=self.__norm(p0+dp)*self.getTolerance() ,
352              rtol=0.,
353              iter_max=max_iter,
354              iter_restart=iter_restart,
355              verbose=self.verbose,P_R=None)
356                
357              u=self.getFlux(p, u0)
358              return u,p
359    
360       def __solve_SYMSTAB(self, u0, p0, max_iter, iter_restart):
361              # p0 is used as an initial guess
362          u=self.getFlux(p0, u0)  
363              self.__pde_p.setValue( Y= self.__f,
364                                     X=  0.5*(self.__g + u - util.tensor_mult(self.__permeability,util.grad(p0)) ),
365                                     y= -   util.inner(self.domain.getNormal(), u),
366                                     r=escript.Data())
367          dp=self.__pde_p.getSolution()
368          p=GMRES(dp,
369                  self.__SYMSTAB_Aprod,
370              p0,
371              self.__inner,
372              atol=self.__norm(p0+dp)*self.getTolerance() ,
373              rtol=0.,
374              iter_max=max_iter,
375              iter_restart=iter_restart,
376              verbose=self.verbose,P_R=None)
377                
378              u=self.getFlux(p, u0)
379              return u,p
380    
381       def __L2(self,v):
382             return util.sqrt(util.integrate(util.length(util.interpolate(v,escript.Function(self.domain)))**2))      
383      
384       def __norm(self,r):
385             return util.sqrt(self.__inner(r,r))
386            
387       def __inner(self,r,s):
388             return util.integrate(util.inner(r,s), escript.Function(self.domain))
389            
390       def __STAB_Aprod(self,p):
391          gp=util.grad(p)
392          self.__pde_v.setValue(Y=-0.5*gp,
393                                X=-p*util.kronecker(self.__pde_v.getDomain()),
394                                y= p * self.domain.getNormal(),  
395                                r=escript.Data())
396          u = -self.__pde_v.getSolution()
397          self.__pde_p.setValue(Y=util.div(u),
398                                X=0.5*(u+util.tensor_mult(self.__permeability,gp)),
399                                y=escript.Data(),
400                                r=escript.Data())
401        
402          return  self.__pde_p.getSolution()
403      
404       def __SYMSTAB_Aprod(self,p):
405          gp=util.grad(p)
406          self.__pde_v.setValue(Y=0.5*gp ,
407                                X=escript.Data(),
408                                y=escript.Data(),  
409                                r=escript.Data())
410          u = -self.__pde_v.getSolution()
411          self.__pde_p.setValue(Y=escript.Data(),
412                                X=0.5*(-u+util.tensor_mult(self.__permeability,gp)),
413                                y=   util.inner(self.domain.getNormal(), u),
414                                r=escript.Data())
415        
416          return  self.__pde_p.getSolution()
417          
418    
419  class StokesProblemCartesian(HomogeneousSaddlePointProblem):  class StokesProblemCartesian(HomogeneousSaddlePointProblem):
420       """       """
# Line 380  class StokesProblemCartesian(Homogeneous Line 435  class StokesProblemCartesian(Homogeneous
435              sp.initialize(...)              sp.initialize(...)
436              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
437       """       """
438       def __init__(self,domain,adaptSubTolerance=True, **kwargs):       def __init__(self,domain,**kwargs):
439           """           """
440           initialize the Stokes Problem           initialize the Stokes Problem
441    
442           @param domain: domain of the problem. The approximation order needs to be two.           The approximation spaces used for velocity (=Solution(domain)) and pressure (=ReducedSolution(domain)) must be
443           @type domain: L{Domain}           LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
444       @param adaptSubTolerance: If True the tolerance for subproblem is set automatically.           with macro elements for the pressure.
445       @type adaptSubTolerance: C{bool}  
446           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.           :param domain: domain of the problem.
447             :type domain: `Domain`
448           """           """
449           HomogeneousSaddlePointProblem.__init__(self,adaptSubTolerance=adaptSubTolerance,**kwargs)           HomogeneousSaddlePointProblem.__init__(self,**kwargs)
450           self.domain=domain           self.domain=domain
451           self.vol=util.integrate(1.,Function(self.domain))           self.__pde_v=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
452           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_v.setSymmetryOn()
          self.__pde_u.setSymmetryOn()  
453            
454           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
455           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
# Line 409  class StokesProblemCartesian(Homogeneous Line 464  class StokesProblemCartesian(Homogeneous
464           """           """
465       returns the solver options used  solve the equation for velocity.       returns the solver options used  solve the equation for velocity.
466            
467       @rtype: L{SolverOptions}       :rtype: `SolverOptions`
468       """       """
469       return self.__pde_u.getSolverOptions()       return self.__pde_v.getSolverOptions()
470       def setSolverOptionsVelocity(self, options=None):       def setSolverOptionsVelocity(self, options=None):
471           """           """
472       set the solver options for solving the equation for velocity.       set the solver options for solving the equation for velocity.
473            
474       @param options: new solver  options       :param options: new solver  options
475       @type options: L{SolverOptions}       :type options: `SolverOptions`
476       """       """
477           self.__pde_u.setSolverOptions(options)           self.__pde_v.setSolverOptions(options)
478       def getSolverOptionsPressure(self):       def getSolverOptionsPressure(self):
479           """           """
480       returns the solver options used  solve the equation for pressure.       returns the solver options used  solve the equation for pressure.
481       @rtype: L{SolverOptions}       :rtype: `SolverOptions`
482       """       """
483       return self.__pde_prec.getSolverOptions()       return self.__pde_prec.getSolverOptions()
484       def setSolverOptionsPressure(self, options=None):       def setSolverOptionsPressure(self, options=None):
485           """           """
486       set the solver options for solving the equation for pressure.       set the solver options for solving the equation for pressure.
487       @param options: new solver  options       :param options: new solver  options
488       @type options: L{SolverOptions}       :type options: `SolverOptions`
489       """       """
490       self.__pde_prec.setSolverOptions(options)       self.__pde_prec.setSolverOptions(options)
491    
# Line 439  class StokesProblemCartesian(Homogeneous Line 494  class StokesProblemCartesian(Homogeneous
494       set the solver options for solving the equation to project the divergence of       set the solver options for solving the equation to project the divergence of
495       the velocity onto the function space of presure.       the velocity onto the function space of presure.
496            
497       @param options: new solver options       :param options: new solver options
498       @type options: L{SolverOptions}       :type options: `SolverOptions`
499       """       """
500       self.__pde_prec.setSolverOptions(options)       self.__pde_proj.setSolverOptions(options)
501       def getSolverOptionsDiv(self):       def getSolverOptionsDiv(self):
502           """           """
503       returns the solver options for solving the equation to project the divergence of       returns the solver options for solving the equation to project the divergence of
504       the velocity onto the function space of presure.       the velocity onto the function space of presure.
505            
506       @rtype: L{SolverOptions}       :rtype: `SolverOptions`
507       """       """
508       return self.__pde_prec.getSolverOptions()       return self.__pde_proj.getSolverOptions()
509       def setSubProblemTolerance(self):  
510         def updateStokesEquation(self, v, p):
511           """           """
512       Updates the tolerance for subproblems           updates the Stokes equation to consider dependencies from ``v`` and ``p``
513             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values.
514           """           """
515       if self.adaptSubTolerance():           pass
516               sub_tol=self.getSubProblemTolerance()       def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
517           self.getSolverOptionsDiv().setTolerance(sub_tol)          """
518           self.getSolverOptionsDiv().setAbsoluteTolerance(0.)          assigns new values to the model parameters.
519           self.getSolverOptionsPressure().setTolerance(sub_tol)  
520           self.getSolverOptionsPressure().setAbsoluteTolerance(0.)          :param f: external force
521           self.getSolverOptionsVelocity().setTolerance(sub_tol)          :type f: `Vector` object in `FunctionSpace` `Function` or similar
522           self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)          :param fixed_u_mask: mask of locations with fixed velocity.
523                    :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
524            :param eta: viscosity
525            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
526            :param surface_stress: normal surface stress
527            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
528            :param stress: initial stress
529        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
530            """
531            if eta !=None:
532                k=util.kronecker(self.domain.getDim())
533                kk=util.outer(k,k)
534                self.eta=util.interpolate(eta, escript.Function(self.domain))
535            self.__pde_prec.setValue(D=1/self.eta)
536                self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
537            if restoration_factor!=None:
538                n=self.domain.getNormal()
539                self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
540            if fixed_u_mask!=None:
541                self.__pde_v.setValue(q=fixed_u_mask)
542            if f!=None: self.__f=f
543            if surface_stress!=None: self.__surface_stress=surface_stress
544            if stress!=None: self.__stress=stress
545    
546       def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):       def initialize(self,f=escript.Data(),fixed_u_mask=escript.Data(),eta=1,surface_stress=escript.Data(),stress=escript.Data(), restoration_factor=0):
547          """          """
548          assigns values to the model parameters          assigns values to the model parameters
549    
550          @param f: external force          :param f: external force
551          @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar          :type f: `Vector` object in `FunctionSpace` `Function` or similar
552          @param fixed_u_mask: mask of locations with fixed velocity.          :param fixed_u_mask: mask of locations with fixed velocity.
553          @type fixed_u_mask: L{Vector} object on L{FunctionSpace} L{Solution} or similar          :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
554          @param eta: viscosity          :param eta: viscosity
555          @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar          :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
556          @param surface_stress: normal surface stress          :param surface_stress: normal surface stress
557          @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar          :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
558          @param stress: initial stress          :param stress: initial stress
559      @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar      :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
         @note: All values needs to be set.  
560          """          """
561          self.eta=eta          self.setStokesEquation(f,fixed_u_mask, eta, surface_stress, stress, restoration_factor)
         A =self.__pde_u.createCoefficient("A")  
     self.__pde_u.setValue(A=Data())  
         for i in range(self.domain.getDim()):  
         for j in range(self.domain.getDim()):  
             A[i,j,j,i] += 1.  
             A[i,j,i,j] += 1.  
     self.__pde_prec.setValue(D=1/self.eta)  
         self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask)  
         self.__f=f  
         self.__surface_stress=surface_stress  
         self.__stress=stress  
562    
563       def Bv(self,v):       def Bv(self,v,tol):
564           """           """
565           returns inner product of element p and div(v)           returns inner product of element p and div(v)
566    
567           @param p: a pressure increment           :param v: a residual
568           @param v: a residual           :return: inner product of element p and div(v)
569           @return: inner product of element p and div(v)           :rtype: ``float``
570           @rtype: C{float}           """
571           """           self.__pde_proj.setValue(Y=-util.div(v))
572           self.__pde_proj.setValue(Y=-util.div(v))       self.getSolverOptionsDiv().setTolerance(tol)
573           return self.__pde_proj.getSolution()       self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
574             out=self.__pde_proj.getSolution()
575             return out
576    
577       def inner_pBv(self,p,Bv):       def inner_pBv(self,p,Bv):
578           """           """
579           returns inner product of element p and Bv=-div(v)           returns inner product of element p and Bv=-div(v)
580    
581           @param p: a pressure increment           :param p: a pressure increment
582           @param v: a residual           :param Bv: a residual
583           @return: inner product of element p and Bv=-div(v)           :return: inner product of element p and Bv=-div(v)
584           @rtype: C{float}           :rtype: ``float``
585           """           """
586           return util.integrate(util.interpolate(p,Function(self.domain))*util.interpolate(Bv,Function(self.domain)))           return util.integrate(util.interpolate(p,escript.Function(self.domain))*util.interpolate(Bv, escript.Function(self.domain)))
587    
588       def inner_p(self,p0,p1):       def inner_p(self,p0,p1):
589           """           """
590           Returns inner product of p0 and p1           Returns inner product of p0 and p1
591    
592           @param p0: a pressure           :param p0: a pressure
593           @param p1: a pressure           :param p1: a pressure
594           @return: inner product of p0 and p1           :return: inner product of p0 and p1
595           @rtype: C{float}           :rtype: ``float``
596           """           """
597           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0, escript.Function(self.domain))
598           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1, escript.Function(self.domain))
599           return util.integrate(s0*s1)           return util.integrate(s0*s1)
600    
601       def norm_v(self,v):       def norm_v(self,v):
602           """           """
603           returns the norm of v           returns the norm of v
604    
605           @param v: a velovity           :param v: a velovity
606           @return: norm of v           :return: norm of v
607           @rtype: non-negative C{float}           :rtype: non-negative ``float``
608           """           """
609           return util.sqrt(util.integrate(util.length(util.grad(v))))           return util.sqrt(util.integrate(util.length(util.grad(v))**2))
610    
611       def getV(self, p, v0):  
612         def getDV(self, p, v, tol):
613           """           """
614           return the value for v for a given p (overwrite)           return the value for v for a given p (overwrite)
615    
616           @param p: a pressure           :param p: a pressure
617           @param v0: a initial guess for the value v to return.           :param v: a initial guess for the value v to return.
618           @return: v given as M{v= A^{-1} (f-B^*p)}           :return: dv given as *Adv=(f-Av-B^*p)*
619           """           """
620           self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress, r=v0)           self.updateStokesEquation(v,p)
621             self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
622         self.getSolverOptionsVelocity().setTolerance(tol)
623         self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
624           if self.__stress.isEmpty():           if self.__stress.isEmpty():
625              self.__pde_u.setValue(X=p*util.kronecker(self.domain))              self.__pde_v.setValue(X=p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
626           else:           else:
627              self.__pde_u.setValue(X=self.__stress+p*util.kronecker(self.domain))              self.__pde_v.setValue(X=self.__stress+p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
628           out=self.__pde_u.getSolution()           out=self.__pde_v.getSolution()
629           return  out           return  out
630    
631       def norm_Bv(self,Bv):       def norm_Bv(self,Bv):
632          """          """
633          Returns Bv (overwrite).          Returns Bv (overwrite).
634    
635          @rtype: equal to the type of p          :rtype: equal to the type of p
636          @note: boundary conditions on p should be zero!          :note: boundary conditions on p should be zero!
637          """          """
638          return util.sqrt(util.integrate(util.interpolate(Bv,Function(self.domain))**2))          return util.sqrt(util.integrate(util.interpolate(Bv, escript.Function(self.domain))**2))
639    
640       def solve_AinvBt(self,p):       def solve_AinvBt(self,p, tol):
641           """           """
642           Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}           Solves *Av=B^*p* with accuracy `tol`
643    
644           @param p: a pressure increment           :param p: a pressure increment
645           @return: the solution of M{Av=B^*p}           :return: the solution of *Av=B^*p*
646           @note: boundary conditions on v should be zero!           :note: boundary conditions on v should be zero!
647           """           """
648           self.__pde_u.setValue(Y=Data(), y=Data(), r=Data(),X=-p*util.kronecker(self.domain))           self.__pde_v.setValue(Y=escript.Data(), y=escript.Data(), X=-p*util.kronecker(self.domain))
649           out=self.__pde_u.getSolution()           out=self.__pde_v.getSolution()
650           return  out           return  out
651    
652       def solve_prec(self,Bv):       def solve_prec(self,Bv, tol):
653           """           """
654           applies preconditioner for for M{BA^{-1}B^*} to M{Bv}           applies preconditioner for for *BA^{-1}B^** to *Bv*
655           with accuracy L{self.getSubProblemTolerance()}           with accuracy `self.getSubProblemTolerance()`
656    
657           @param v: velocity increment           :param Bv: velocity increment
658           @return: M{p=P(Bv)} where M{P^{-1}} is an approximation of M{BA^{-1}B^*}           :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
659           @note: boundary conditions on p are zero.           :note: boundary conditions on p are zero.
660           """           """
661           self.__pde_prec.setValue(Y=Bv)           self.__pde_prec.setValue(Y=Bv)
662           return self.__pde_prec.getSolution()       self.getSolverOptionsPressure().setTolerance(tol)
663         self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
664             out=self.__pde_prec.getSolution()
665             return out

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