/[escript]/trunk/escript/py_src/flows.py
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revision 1562 by gross, Wed May 21 13:04:40 2008 UTC revision 2960 by gross, Tue Mar 2 07:54:11 2010 UTC
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1  # $Id:$  ########################################################
2  #  #
3  #######################################################  # Copyright (c) 2003-2010 by University of Queensland
4    # Earth Systems Science Computational Center (ESSCC)
5    # http://www.uq.edu.au/esscc
6  #  #
7  #       Copyright 2008 by University of Queensland  # Primary Business: Queensland, Australia
8  #  # Licensed under the Open Software License version 3.0
9  #                http://esscc.uq.edu.au  # http://www.opensource.org/licenses/osl-3.0.php
 #        Primary Business: Queensland, Australia  
 #  Licensed under the Open Software License version 3.0  
 #     http://www.opensource.org/licenses/osl-3.0.php  
 #  
 #######################################################  
10  #  #
11    ########################################################
12    
13    __copyright__="""Copyright (c) 2003-2010 by University of Queensland
14    Earth Systems Science Computational Center (ESSCC)
15    http://www.uq.edu.au/esscc
16    Primary Business: Queensland, Australia"""
17    __license__="""Licensed under the Open Software License version 3.0
18    http://www.opensource.org/licenses/osl-3.0.php"""
19    __url__="https://launchpad.net/escript-finley"
20    
21  """  """
22  Some models for flow  Some models for flow
23    
24  @var __author__: name of author  :var __author__: name of author
25  @var __copyright__: copyrights  :var __copyright__: copyrights
26  @var __license__: licence agreement  :var __license__: licence agreement
27  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
28  @var __version__: version  :var __version__: version
29  @var __date__: date of the version  :var __date__: date of the version
30  """  """
31    
32  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
 __copyright__="""  Copyright (c) 2008 by ACcESS MNRF  
                     http://www.access.edu.au  
                 Primary Business: Queensland, Australia"""  
 __license__="""Licensed under the Open Software License version 3.0  
              http://www.opensource.org/licenses/osl-3.0.php"""  
 __url__="http://www.iservo.edu.au/esys"  
 __version__="$Revision:$"  
 __date__="$Date:$"  
33    
34  from escript import *  from escript import *
35  import util  import util
36  from linearPDEs import LinearPDE  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
37  from pdetools import HomogeneousSaddlePointProblem,Projector  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
38    
39    class DarcyFlow(object):
40        """
41        solves the problem
42    
43        *u_i+k_{ij}*p_{,j} = g_i*
44        *u_{i,i} = f*
45    
46        where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
47    
48        :note: The problem is solved in a least squares formulation.
49        """
50    
51        def __init__(self, domain, weight=None, useReduced=False, adaptSubTolerance=True):
52            """
53            initializes the Darcy flux problem
54            :param domain: domain of the problem
55            :type domain: `Domain`
56            :param weight: the weighting factor for the incempressiblity equation *u_{i,i} = f* in the variational principle.
57            If not present an apprppriate weight is chosen.
58            :type  weight: positive sacalar
59        :param useReduced: uses reduced oreder on flux and pressure
60        :type useReduced: ``bool``
61        :param adaptSubTolerance: switches on automatic subtolerance selection
62        :type adaptSubTolerance: ``bool``  
63            """
64            self.domain=domain
65            if weight == None:
66               self.__l=None
67               self.__update_weight=True
68            else:
69               self.__update_weight=False
70               self.__l=weight
71            self.__pde_v=LinearPDESystem(domain)
72            if useReduced: self.__pde_v.setReducedOrderOn()
73            self.__pde_v.setSymmetryOn()
74            self.__pde_p=LinearSinglePDE(domain)
75            self.__pde_p.setSymmetryOn()
76            if useReduced: self.__pde_p.setReducedOrderOn()
77            self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
78            self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
79            self.setTolerance()
80            self.setAbsoluteTolerance()
81        self.__adaptSubTolerance=adaptSubTolerance
82        self.verbose=False
83        def getSolverOptionsFlux(self):
84        """
85        Returns the solver options used to solve the flux problems
86        
87        *(I+D^* weight D)u=F*
88        
89        :return: `SolverOptions`
90        """
91        return self.__pde_v.getSolverOptions()
92    
93        def setSolverOptionsFlux(self, options=None):
94        """
95        Sets the solver options used to solve the flux problems
96        
97        *(I+D^*  weight  D)u=F*
98        
99        If ``options`` is not present, the options are reset to default
100        :param options: `SolverOptions`
101        :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
102        """
103        return self.__pde_v.setSolverOptions(options)
104        def getSolverOptionsPressure(self):
105        """
106        Returns the solver options used to solve the pressure problems
107        
108        *(Q^* S Q)p=Q^*G*
109        
110             as a preconditioner where S=weight*k^2/dx^2
111    
112        :return: `SolverOptions`
113        """
114        return self.__pde_p.getSolverOptions()
115        def setSolverOptionsPressure(self, options=None):
116        """
117        Sets the solver options used to solve the pressure problems
118        
119        *(Q^* S Q)p=Q^*G*
120        
121            was a preconditioner here S=weight*k^2/dx^2
122        
123        If ``options`` is not present, the options are reset to default
124    
125        :param options: `SolverOptions`
126        :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
127        """
128        return self.__pde_p.setSolverOptions(options)
129    
130        def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
131            """
132            assigns values to model parameters
133    
134            :param f: volumetic sources/sinks
135            :type f: scalar value on the domain (e.g. `Data`)
136            :param g: flux sources/sinks
137            :type g: vector values on the domain (e.g. `Data`)
138            :param location_of_fixed_pressure: mask for locations where pressure is fixed
139            :type location_of_fixed_pressure: scalar value on the domain (e.g. `Data`)
140            :param location_of_fixed_flux:  mask for locations where flux is fixed.
141            :type location_of_fixed_flux: vector values on the domain (e.g. `Data`)
142            :param permeability: permeability tensor. If scalar ``s`` is given the tensor with
143                                 ``s`` on the main diagonal is used. If vector ``v`` is given the tensor with
144                                 ``v`` on the main diagonal is used.
145            :type permeability: scalar, vector or tensor values on the domain (e.g. `Data`)
146    
147            :note: the values of parameters which are not set by calling ``setValue`` are not altered.
148            :note: at any point on the boundary of the domain the pressure (``location_of_fixed_pressure`` >0)
149                   or the normal component of the flux (``location_of_fixed_flux[i]>0`` if direction of the normal
150                   is along the *x_i* axis.
151            """
152            if f !=None:
153               f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
154               if f.isEmpty():
155                   f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
156               else:
157                   if f.getRank()>0: raise ValueError,"illegal rank of f."
158               self.__f=f
159            if g !=None:
160               g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
161               if g.isEmpty():
162                 g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
163               else:
164                 if not g.getShape()==(self.domain.getDim(),):
165                   raise ValueError,"illegal shape of g"
166               self.__g=g
167    
168            if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)
169            if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)
170    
171            if permeability!=None:
172               perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
173               if perm.getRank()==0:
174                   perm_inv=1./perm*util.kronecker(self.domain.getDim())
175                   perm=perm*util.kronecker(self.domain.getDim())
176               elif perm.getRank()==1:
177                   perm2=perm
178                   perm=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A"))
179                   perm_inv=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A"))
180                   for i in range(self.domain.getDim()):
181                          p=perm2[i]
182                          perm[i,i]=p
183                          perm_inv[i,i]=1/p
184               elif perm.getRank()==2:
185                   perm_inv=util.inverse(perm)
186               else:
187                  raise ValueError,"illegal rank of permeability."
188               self.__permeability=perm
189               self.__permeability_inv=perm_inv
190               if self.__update_weight: self.__l = util.longestEdge(self.domain)**2/3.14**2*util.length(self.__permeability_inv)
191               self.__pde_v.setValue(D=self.__permeability_inv, A=self.__l*util.outer(util.kronecker(self.domain),util.kronecker(self.domain)))
192               s=self.__pde_p.getFunctionSpaceForCoefficient("A").getSize()
193               self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability)*self.__l/s**2)
194    
195        def getFlux(self,p=None, fixed_flux=Data()):
196            """
197            returns the flux for a given pressure ``p`` where the flux is equal to ``fixed_flux``
198            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
199            Note that ``g`` and ``f`` are used, see `setValue`.
200    
201            :param p: pressure.
202            :type p: scalar value on the domain (e.g. `Data`).
203            :param fixed_flux: flux on the locations of the domain marked be ``location_of_fixed_flux``.
204            :type fixed_flux: vector values on the domain (e.g. `Data`).
205            :return: flux
206            :rtype: `Data`
207            :note: the method uses the least squares solution *u=(I+D^*D)^{-1}(D^*f-g-Qp)* where *D* is the *div* operator and *(Qp)_i=k_{ij}p_{,j}*
208                   for the permeability *k_{ij}*
209            """
210        self.setSubProblemTolerance()
211            self.__pde_v.setValue(X=util.sqrt(self.__l)*self.__f*util.kronecker(self.domain), r=fixed_flux)
212            g2=util.tensor_mult(self.__permeability_inv,self.__g)
213            if p == None:
214               self.__pde_v.setValue(Y=g2)
215            else:
216               self.__pde_v.setValue(Y=g2-util.grad(p))
217            return self.__pde_v.getSolution()
218    
219        def __Aprod(self,dp):
220              if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator"
221              Gdp=util.grad(dp)
222              self.__pde_v.setValue(Y=-Gdp,X=Data(), r=Data())
223              du=self.__pde_v.getSolution()
224              # self.__pde_v.getOperator().saveMM("proj.mm")
225              return ArithmeticTuple(util.tensor_mult(self.__permeability,Gdp),-du)
226    
227        def __Msolve_PCG(self,r):
228          if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner"
229              self.__pde_p.setValue(X=r[0]-r[1], Y=Data(), r=Data())
230              # self.__pde_p.getOperator().saveMM("prec.mm")
231              return self.__pde_p.getSolution()
232    
233        def __inner_PCG(self,p,r):
234             return util.integrate(util.inner(util.grad(p), r[0]-r[1]))
235    
236        def __L2(self,v):
237             return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
238    
239        def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10):
240             """
241             solves the problem.
242    
243             The iteration is terminated if the residual norm is less then self.getTolerance().
244    
245             :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.
246             :type u0: vector value on the domain (e.g. `Data`).
247             :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.
248             :type p0: scalar value on the domain (e.g. `Data`).
249             :param verbose: if set some information on iteration progress are printed
250             :type verbose: ``bool``
251             :return: flux and pressure
252             :rtype: ``tuple`` of `Data`.
253    
254             :note: The problem is solved as a least squares form
255    
256             *(K^[-1]+D^* weight D)u+G p=D^* sqrt(weight) f + K^[-1]g*
257             *G^*u+G^* K Gp=G^*g*
258    
259             where *D* is the *div* operator and *(Gp)_i=p_{,i}* for the permeability *K=k_{ij}*.
260             We eliminate the flux form the problem by setting
261    
262             *u=(K^[-1]+D^* weight D)^{-1}(D^* sqrt(weight) f + K^[-1]g* -G p ) with u=u0 on location_of_fixed_flux
263    
264             form the first equation. Inserted into the second equation we get
265    
266             *G^*(K-(K^[-1]+D^* weight D)^{-1})Gp= G^*(g-(K^[-1]+D^* weight D)^{-1}(D^* sqrt(weight) f + K^[-1]g))* with p=p0  on location_of_fixed_pressure
267    
268             which is solved using the PCG method (precondition is *G^*K^2*weight*l^2/dx^2 G*. In each iteration step
269             PDEs with operator *K^[-1]+D^* weight D* and with *G^*K^2*weight*l^2/dx^2 G* needs to be solved using a sub iteration scheme.
270             """
271             self.verbose=verbose
272             rtol=self.getTolerance()
273             atol=self.getAbsoluteTolerance()
274         self.setSubProblemTolerance()
275             num_corrections=0
276             converged=False
277             p=p0
278             norm_r=None
279             v=self.getFlux(p, fixed_flux=u0)
280             while not converged:
281                   Gp=util.grad(p)
282                   KGp=util.tensor_mult(self.__permeability,Gp)
283                   norm_v=util.integrate(util.inner(v,util.tensor_mult(self.__permeability_inv,v)))**0.5
284                   norm_Gp=util.integrate(util.inner(Gp,KGp))**0.5
285                   if self.verbose:
286                       print "DarcyFlux: L2: g-v-K*grad(p) = %e."%(util.integrate(util.length(self.__g-util.interpolate(v,Function(self.domain))-KGp)**2)**(0.5),)
287                       print "DarcyFlux: L2: f-div(v) = %e."%(util.integrate((self.__f-util.div(v))**2)**(0.5),)
288                       print "DarcyFlux: K^{-1}-norm of v = %e."%norm_v
289                       print "DarcyFlux: K-norm of grad(p) = %e."%norm_Gp
290                   if norm_v == 0.:
291                      if norm_Gp == 0.:
292                         return v,p
293                      else:
294                        fac=norm_Gp
295                   else:
296                      if norm_Gp == 0.:
297                        fac=norm_v
298                      else:
299                        fac=2./(1./norm_v+1./norm_Gp)
300                   #fac=max(norm_Gp, norm_v)
301                   ATOL=atol+rtol*fac
302                   if self.verbose: print "DarcyFlux: absolute tolerance ATOL = %e (fac= %e)."%(ATOL,fac)
303                   if norm_r == None or norm_r>ATOL:
304                       if num_corrections>max_num_corrections:
305                             raise ValueError,"maximum number of correction steps reached."
306                       # initial residual is r=G^*(self.__g-KGp - v)
307                       p,r, norm_r=PCG(ArithmeticTuple(self.__g-KGp,v),
308                                       self.__Aprod,
309                                       p,
310                                       self.__Msolve_PCG,
311                                       self.__inner_PCG,
312                                       atol=ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
313                       if self.verbose: print "DarcyFlux: residual norm = %e."%norm_r
314                       v=r[1]
315                       num_corrections+=1
316                   else:
317                       if self.verbose: print "DarcyFlux: stopping criterium reached."
318                       converged=True
319             return v,p
320    
321              
322    
323        def setTolerance(self,rtol=1e-4):
324            """
325            sets the relative tolerance ``rtol`` used to terminate the solution process. The iteration is terminated if
326    
327            *|g-v-K gard(p)|_PCG <= atol + rtol * (1/|K^{-1/2}u|_0 +1/|K^{1/2}grad(p)|_0)^(-1}*
328    
329            where ``atol`` is an absolut tolerance (see `setAbsoluteTolerance`).
330    
331            :param rtol: relative tolerance for the pressure
332            :type rtol: non-negative ``float``
333            """
334            if rtol<0:
335                raise ValueError,"Relative tolerance needs to be non-negative."
336            self.__rtol=rtol
337        def getTolerance(self):
338            """
339            returns the relative tolerance
340    
341            :return: current relative tolerance
342            :rtype: ``float``
343            """
344            return self.__rtol
345    
346        def setAbsoluteTolerance(self,atol=0.):
347            """
348            sets the absolute tolerance ``atol`` used to terminate the solution process. The iteration is terminated if
349    
350             *|g-v-K gard(p)|_PCG <= atol + rtol * (1/|K^{-1/2}u|_0 +1/|K^{1/2}grad(p)|_0)^(-1}*
351    
352            where ``rtol`` is an absolut tolerance (see `setTolerance`), *|f|^2 = integrate(length(f)^2)* and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*.
353    
354            :param atol: absolute tolerance for the pressure
355            :type atol: non-negative ``float``
356            """
357            if atol<0:
358                raise ValueError,"Absolute tolerance needs to be non-negative."
359            self.__atol=atol
360        def getAbsoluteTolerance(self):
361           """
362           returns the absolute tolerance
363          
364           :return: current absolute tolerance
365           :rtype: ``float``
366           """
367           return self.__atol
368        def getSubProblemTolerance(self):
369        """
370        Returns a suitable subtolerance
371        @type: ``float``
372        """
373        return max(util.EPSILON**(0.75),self.getTolerance()**2)
374    
375        def setSubProblemTolerance(self):
376             """
377             Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected.
378             """
379         if self.__adaptSubTolerance:
380             sub_tol=self.getSubProblemTolerance()
381                 self.getSolverOptionsFlux().setTolerance(sub_tol)
382             self.getSolverOptionsFlux().setAbsoluteTolerance(0.)
383             self.getSolverOptionsPressure().setTolerance(sub_tol)
384             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
385             if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol
386    
387    
388    class DarcyFlowOld(object):
389        """
390        solves the problem
391    
392        *u_i+k_{ij}*p_{,j} = g_i*
393        *u_{i,i} = f*
394    
395        where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
396    
397        :note: The problem is solved in a least squares formulation.
398        """
399    
400        def __init__(self, domain, weight=None, useReduced=False, adaptSubTolerance=True):
401            """
402            initializes the Darcy flux problem
403            :param domain: domain of the problem
404            :type domain: `Domain`
405        :param useReduced: uses reduced oreder on flux and pressure
406        :type useReduced: ``bool``
407        :param adaptSubTolerance: switches on automatic subtolerance selection
408        :type adaptSubTolerance: ``bool``  
409            """
410            self.domain=domain
411            if weight == None:
412               s=self.domain.getSize()
413               self.__l=(3.*util.longestEdge(self.domain)*s/util.sup(s))**2
414               # self.__l=(3.*util.longestEdge(self.domain))**2
415               #self.__l=(0.1*util.longestEdge(self.domain)*s/util.sup(s))**2
416            else:
417               self.__l=weight
418            self.__pde_v=LinearPDESystem(domain)
419            if useReduced: self.__pde_v.setReducedOrderOn()
420            self.__pde_v.setSymmetryOn()
421            self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain)))
422            self.__pde_p=LinearSinglePDE(domain)
423            self.__pde_p.setSymmetryOn()
424            if useReduced: self.__pde_p.setReducedOrderOn()
425            self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
426            self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
427            self.setTolerance()
428            self.setAbsoluteTolerance()
429        self.__adaptSubTolerance=adaptSubTolerance
430        self.verbose=False
431        def getSolverOptionsFlux(self):
432        """
433        Returns the solver options used to solve the flux problems
434        
435        *(I+D^*D)u=F*
436        
437        :return: `SolverOptions`
438        """
439        return self.__pde_v.getSolverOptions()
440        def setSolverOptionsFlux(self, options=None):
441        """
442        Sets the solver options used to solve the flux problems
443        
444        *(I+D^*D)u=F*
445        
446        If ``options`` is not present, the options are reset to default
447        :param options: `SolverOptions`
448        :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
449        """
450        return self.__pde_v.setSolverOptions(options)
451        def getSolverOptionsPressure(self):
452        """
453        Returns the solver options used to solve the pressure problems
454        
455        *(Q^*Q)p=Q^*G*
456        
457        :return: `SolverOptions`
458        """
459        return self.__pde_p.getSolverOptions()
460        def setSolverOptionsPressure(self, options=None):
461        """
462        Sets the solver options used to solve the pressure problems
463        
464        *(Q^*Q)p=Q^*G*
465        
466        If ``options`` is not present, the options are reset to default
467        :param options: `SolverOptions`
468        :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
469        """
470        return self.__pde_p.setSolverOptions(options)
471    
472        def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
473            """
474            assigns values to model parameters
475    
476            :param f: volumetic sources/sinks
477            :type f: scalar value on the domain (e.g. `Data`)
478            :param g: flux sources/sinks
479            :type g: vector values on the domain (e.g. `Data`)
480            :param location_of_fixed_pressure: mask for locations where pressure is fixed
481            :type location_of_fixed_pressure: scalar value on the domain (e.g. `Data`)
482            :param location_of_fixed_flux:  mask for locations where flux is fixed.
483            :type location_of_fixed_flux: vector values on the domain (e.g. `Data`)
484            :param permeability: permeability tensor. If scalar ``s`` is given the tensor with
485                                 ``s`` on the main diagonal is used. If vector ``v`` is given the tensor with
486                                 ``v`` on the main diagonal is used.
487            :type permeability: scalar, vector or tensor values on the domain (e.g. `Data`)
488    
489            :note: the values of parameters which are not set by calling ``setValue`` are not altered.
490            :note: at any point on the boundary of the domain the pressure (``location_of_fixed_pressure`` >0)
491                   or the normal component of the flux (``location_of_fixed_flux[i]>0`` if direction of the normal
492                   is along the *x_i* axis.
493            """
494            if f !=None:
495               f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
496               if f.isEmpty():
497                   f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
498               else:
499                   if f.getRank()>0: raise ValueError,"illegal rank of f."
500               self.__f=f
501            if g !=None:
502               g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
503               if g.isEmpty():
504                 g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
505               else:
506                 if not g.getShape()==(self.domain.getDim(),):
507                   raise ValueError,"illegal shape of g"
508               self.__g=g
509    
510            if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)
511            if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)
512    
513            if permeability!=None:
514               perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
515               if perm.getRank()==0:
516                   perm=perm*util.kronecker(self.domain.getDim())
517               elif perm.getRank()==1:
518                   perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm
519                   for i in range(self.domain.getDim()): perm[i,i]=perm2[i]
520               elif perm.getRank()==2:
521                  pass
522               else:
523                  raise ValueError,"illegal rank of permeability."
524               self.__permeability=perm
525               self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))
526    
527        def setTolerance(self,rtol=1e-4):
528            """
529            sets the relative tolerance ``rtol`` used to terminate the solution process. The iteration is terminated if
530    
531            *|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) )*
532    
533            where ``atol`` is an absolut tolerance (see `setAbsoluteTolerance`), *|f|^2 = integrate(length(f)^2)* and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*.
534    
535            :param rtol: relative tolerance for the pressure
536            :type rtol: non-negative ``float``
537            """
538            if rtol<0:
539                raise ValueError,"Relative tolerance needs to be non-negative."
540            self.__rtol=rtol
541        def getTolerance(self):
542            """
543            returns the relative tolerance
544    
545            :return: current relative tolerance
546            :rtype: ``float``
547            """
548            return self.__rtol
549    
550        def setAbsoluteTolerance(self,atol=0.):
551            """
552            sets the absolute tolerance ``atol`` used to terminate the solution process. The iteration is terminated if
553    
554            *|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) )*
555    
556            where ``rtol`` is an absolut tolerance (see `setTolerance`), *|f|^2 = integrate(length(f)^2)* and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*.
557    
558            :param atol: absolute tolerance for the pressure
559            :type atol: non-negative ``float``
560            """
561            if atol<0:
562                raise ValueError,"Absolute tolerance needs to be non-negative."
563            self.__atol=atol
564        def getAbsoluteTolerance(self):
565           """
566           returns the absolute tolerance
567          
568           :return: current absolute tolerance
569           :rtype: ``float``
570           """
571           return self.__atol
572        def getSubProblemTolerance(self):
573        """
574        Returns a suitable subtolerance
575        @type: ``float``
576        """
577        return max(util.EPSILON**(0.75),self.getTolerance()**2)
578        def setSubProblemTolerance(self):
579             """
580             Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected.
581             """
582         if self.__adaptSubTolerance:
583             sub_tol=self.getSubProblemTolerance()
584                 self.getSolverOptionsFlux().setTolerance(sub_tol)
585             self.getSolverOptionsFlux().setAbsoluteTolerance(0.)
586             self.getSolverOptionsPressure().setTolerance(sub_tol)
587             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
588             if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol
589    
590        def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10):
591             """
592             solves the problem.
593    
594             The iteration is terminated if the residual norm is less then self.getTolerance().
595    
596             :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.
597             :type u0: vector value on the domain (e.g. `Data`).
598             :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.
599             :type p0: scalar value on the domain (e.g. `Data`).
600             :param verbose: if set some information on iteration progress are printed
601             :type verbose: ``bool``
602             :return: flux and pressure
603             :rtype: ``tuple`` of `Data`.
604    
605             :note: The problem is solved as a least squares form
606    
607             *(I+D^*D)u+Qp=D^*f+g*
608             *Q^*u+Q^*Qp=Q^*g*
609    
610             where *D* is the *div* operator and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*.
611             We eliminate the flux form the problem by setting
612    
613             *u=(I+D^*D)^{-1}(D^*f-g-Qp)* with u=u0 on location_of_fixed_flux
614    
615             form the first equation. Inserted into the second equation we get
616    
617             *Q^*(I-(I+D^*D)^{-1})Qp= Q^*(g-(I+D^*D)^{-1}(D^*f+g))* with p=p0  on location_of_fixed_pressure
618    
619             which is solved using the PCG method (precondition is *Q^*Q*). In each iteration step
620             PDEs with operator *I+D^*D* and with *Q^*Q* needs to be solved using a sub iteration scheme.
621             """
622             self.verbose=verbose
623             rtol=self.getTolerance()
624             atol=self.getAbsoluteTolerance()
625         self.setSubProblemTolerance()
626             num_corrections=0
627             converged=False
628             p=p0
629             norm_r=None
630             while not converged:
631                   v=self.getFlux(p, fixed_flux=u0)
632                   Qp=self.__Q(p)
633                   norm_v=self.__L2(v)
634                   norm_Qp=self.__L2(Qp)
635                   if norm_v == 0.:
636                      if norm_Qp == 0.:
637                         return v,p
638                      else:
639                        fac=norm_Qp
640                   else:
641                      if norm_Qp == 0.:
642                        fac=norm_v
643                      else:
644                        fac=2./(1./norm_v+1./norm_Qp)
645                   ATOL=(atol+rtol*fac)
646                   if self.verbose:
647                        print "DarcyFlux: L2 norm of v = %e."%norm_v
648                        print "DarcyFlux: L2 norm of k.util.grad(p) = %e."%norm_Qp
649                        print "DarcyFlux: L2 defect u = %e."%(util.integrate(util.length(self.__g-util.interpolate(v,Function(self.domain))-Qp)**2)**(0.5),)
650                        print "DarcyFlux: L2 defect div(v) = %e."%(util.integrate((self.__f-util.div(v))**2)**(0.5),)
651                        print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
652                   if norm_r == None or norm_r>ATOL:
653                       if num_corrections>max_num_corrections:
654                             raise ValueError,"maximum number of correction steps reached."
655                       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)
656                       num_corrections+=1
657                   else:
658                       converged=True
659             return v,p
660        def __L2(self,v):
661             return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
662    
663        def __Q(self,p):
664              return util.tensor_mult(self.__permeability,util.grad(p))
665    
666        def __Aprod(self,dp):
667              if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator"
668              Qdp=self.__Q(dp)
669              self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())
670              du=self.__pde_v.getSolution()
671              # self.__pde_v.getOperator().saveMM("proj.mm")
672              return Qdp+du
673        def __inner_GMRES(self,r,s):
674             return util.integrate(util.inner(r,s))
675    
676        def __inner_PCG(self,p,r):
677             return util.integrate(util.inner(self.__Q(p), r))
678    
679        def __Msolve_PCG(self,r):
680          if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner"
681              self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data())
682              # self.__pde_p.getOperator().saveMM("prec.mm")
683              return self.__pde_p.getSolution()
684    
685        def getFlux(self,p=None, fixed_flux=Data()):
686            """
687            returns the flux for a given pressure ``p`` where the flux is equal to ``fixed_flux``
688            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
689            Note that ``g`` and ``f`` are used, see `setValue`.
690    
691            :param p: pressure.
692            :type p: scalar value on the domain (e.g. `Data`).
693            :param fixed_flux: flux on the locations of the domain marked be ``location_of_fixed_flux``.
694            :type fixed_flux: vector values on the domain (e.g. `Data`).
695            :return: flux
696            :rtype: `Data`
697            :note: the method uses the least squares solution *u=(I+D^*D)^{-1}(D^*f-g-Qp)* where *D* is the *div* operator and *(Qp)_i=k_{ij}p_{,j}*
698                   for the permeability *k_{ij}*
699            """
700        self.setSubProblemTolerance()
701            g=self.__g
702            f=self.__f
703            self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)
704            if p == None:
705               self.__pde_v.setValue(Y=g)
706            else:
707               self.__pde_v.setValue(Y=g-self.__Q(p))
708            return self.__pde_v.getSolution()
709    
710  class StokesProblemCartesian(HomogeneousSaddlePointProblem):  class StokesProblemCartesian(HomogeneousSaddlePointProblem):
711        """       """
712        solves       solves
713    
714            -(eta*(u_{i,j}+u_{j,i}))_j - p_i = f_i            -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
715                  u_{i,i}=0                  u_{i,i}=0
716    
717            u=0 where  fixed_u_mask>0            u=0 where  fixed_u_mask>0
718            eta*(u_{i,j}+u_{j,i})*n_j=surface_stress            eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
719    
720        if surface_stress is not give 0 is assumed.       if surface_stress is not given 0 is assumed.
721    
722        typical usage:       typical usage:
723    
724              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
725              sp.setTolerance()              sp.setTolerance()
726              sp.initialize(...)              sp.initialize(...)
727              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
728        """       """
729        def __init__(self,domain,**kwargs):       def __init__(self,domain,**kwargs):
730             """
731             initialize the Stokes Problem
732    
733             The approximation spaces used for velocity (=Solution(domain)) and pressure (=ReducedSolution(domain)) must be
734             LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
735             with macro elements for the pressure.
736    
737             :param domain: domain of the problem.
738             :type domain: `Domain`
739             """
740           HomogeneousSaddlePointProblem.__init__(self,**kwargs)           HomogeneousSaddlePointProblem.__init__(self,**kwargs)
741           self.domain=domain           self.domain=domain
          self.vol=util.integrate(1.,Function(self.domain))  
742           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
743           self.__pde_u.setSymmetryOn()           self.__pde_u.setSymmetryOn()
744           self.__pde_u.setSolverMethod(preconditioner=LinearPDE.ILU0)      
               
745           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
746           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
747           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
748    
749           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=LinearPDE(domain)
750           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
751         self.__pde_proj.setValue(D=1)
752           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
          self.__pde_proj.setValue(D=1.)  
753    
754        def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data()):       def getSolverOptionsVelocity(self):
755          self.eta=eta           """
756          A =self.__pde_u.createCoefficientOfGeneralPDE("A")       returns the solver options used  solve the equation for velocity.
757      self.__pde_u.setValue(A=Data())      
758          for i in range(self.domain.getDim()):       :rtype: `SolverOptions`
759          for j in range(self.domain.getDim()):       """
760              A[i,j,j,i] += 1.       return self.__pde_u.getSolverOptions()
761              A[i,j,i,j] += 1.       def setSolverOptionsVelocity(self, options=None):
762      self.__pde_prec.setValue(D=1/self.eta)           """
763          self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask,Y=f,y=surface_stress)       set the solver options for solving the equation for velocity.
764        
765        def B(self,arg):       :param options: new solver  options
766           d=util.div(arg)       :type options: `SolverOptions`
767           self.__pde_proj.setValue(Y=d)       """
768           self.__pde_proj.setTolerance(self.getSubProblemTolerance())           self.__pde_u.setSolverOptions(options)
769           return self.__pde_proj.getSolution(verbose=self.show_details)       def getSolverOptionsPressure(self):
770             """
771         returns the solver options used  solve the equation for pressure.
772         :rtype: `SolverOptions`
773         """
774         return self.__pde_prec.getSolverOptions()
775         def setSolverOptionsPressure(self, options=None):
776             """
777         set the solver options for solving the equation for pressure.
778         :param options: new solver  options
779         :type options: `SolverOptions`
780         """
781         self.__pde_prec.setSolverOptions(options)
782    
783         def setSolverOptionsDiv(self, options=None):
784             """
785         set the solver options for solving the equation to project the divergence of
786         the velocity onto the function space of presure.
787        
788         :param options: new solver options
789         :type options: `SolverOptions`
790         """
791         self.__pde_proj.setSolverOptions(options)
792         def getSolverOptionsDiv(self):
793             """
794         returns the solver options for solving the equation to project the divergence of
795         the velocity onto the function space of presure.
796        
797         :rtype: `SolverOptions`
798         """
799         return self.__pde_proj.getSolverOptions()
800    
801         def updateStokesEquation(self, v, p):
802             """
803             updates the Stokes equation to consider dependencies from ``v`` and ``p``
804             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values.
805             """
806             pass
807         def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
808            """
809            assigns new values to the model parameters.
810    
811            :param f: external force
812            :type f: `Vector` object in `FunctionSpace` `Function` or similar
813            :param fixed_u_mask: mask of locations with fixed velocity.
814            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
815            :param eta: viscosity
816            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
817            :param surface_stress: normal surface stress
818            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
819            :param stress: initial stress
820        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
821            """
822            if eta !=None:
823                k=util.kronecker(self.domain.getDim())
824                kk=util.outer(k,k)
825                self.eta=util.interpolate(eta, Function(self.domain))
826            self.__pde_prec.setValue(D=1/self.eta)
827                self.__pde_u.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
828            if restoration_factor!=None:
829                n=self.domain.getNormal()
830                self.__pde_u.setValue(d=restoration_factor*util.outer(n,n))
831            if fixed_u_mask!=None:
832                self.__pde_u.setValue(q=fixed_u_mask)
833            if f!=None: self.__f=f
834            if surface_stress!=None: self.__surface_stress=surface_stress
835            if stress!=None: self.__stress=stress
836    
837         def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data(), restoration_factor=0):
838            """
839            assigns values to the model parameters
840    
841            :param f: external force
842            :type f: `Vector` object in `FunctionSpace` `Function` or similar
843            :param fixed_u_mask: mask of locations with fixed velocity.
844            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
845            :param eta: viscosity
846            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
847            :param surface_stress: normal surface stress
848            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
849            :param stress: initial stress
850        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
851            """
852            self.setStokesEquation(f,fixed_u_mask, eta, surface_stress, stress, restoration_factor)
853    
854         def Bv(self,v,tol):
855             """
856             returns inner product of element p and div(v)
857    
858             :param v: a residual
859             :return: inner product of element p and div(v)
860             :rtype: ``float``
861             """
862             self.__pde_proj.setValue(Y=-util.div(v))
863         self.getSolverOptionsDiv().setTolerance(tol)
864         self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
865             out=self.__pde_proj.getSolution()
866             return out
867    
868         def inner_pBv(self,p,Bv):
869             """
870             returns inner product of element p and Bv=-div(v)
871    
872             :param p: a pressure increment
873             :param Bv: a residual
874             :return: inner product of element p and Bv=-div(v)
875             :rtype: ``float``
876             """
877             return util.integrate(util.interpolate(p,Function(self.domain))*util.interpolate(Bv,Function(self.domain)))
878    
879        def inner(self,p0,p1):       def inner_p(self,p0,p1):
880             """
881             Returns inner product of p0 and p1
882    
883             :param p0: a pressure
884             :param p1: a pressure
885             :return: inner product of p0 and p1
886             :rtype: ``float``
887             """
888           s0=util.interpolate(p0,Function(self.domain))           s0=util.interpolate(p0,Function(self.domain))
889           s1=util.interpolate(p1,Function(self.domain))           s1=util.interpolate(p1,Function(self.domain))
890           return util.integrate(s0*s1)           return util.integrate(s0*s1)
891    
892        def inner_a(self,a0,a1):       def norm_v(self,v):
893           p0=util.interpolate(a0[1],Function(self.domain))           """
894           p1=util.interpolate(a1[1],Function(self.domain))           returns the norm of v
895           alfa=(1/self.vol)*util.integrate(p0)  
896           beta=(1/self.vol)*util.integrate(p1)           :param v: a velovity
897       v0=util.grad(a0[0])           :return: norm of v
898       v1=util.grad(a1[0])           :rtype: non-negative ``float``
899       #print "NORM",alfa,beta,util.integrate((p0-alfa)*(p1-beta))+util.integrate(util.inner(v0,v1))           """
900           return util.integrate((p0-alfa)*(p1-beta)+((1/self.eta)**2)*util.inner(v0,v1))           return util.sqrt(util.integrate(util.length(util.grad(v))**2))
   
   
       def getStress(self,u):  
          mg=util.grad(u)  
          return 2.*self.eta*util.symmetric(mg)  
   
       def solve_A(self,u,p):  
          """  
          solves Av=f-Au-B^*p (v=0 on fixed_u_mask)  
          """  
          self.__pde_u.setTolerance(self.getSubProblemTolerance())  
          self.__pde_u.setValue(X=-self.getStress(u)-p*util.kronecker(self.domain))  
          return  self.__pde_u.getSolution(verbose=self.show_details)  
   
   
       def solve_prec(self,p):  
      #proj=Projector(domain=self.domain, reduce = True, fast=False)  
          self.__pde_prec.setTolerance(self.getSubProblemTolerance())  
          self.__pde_prec.setValue(Y=p)  
          q=self.__pde_prec.getSolution(verbose=self.show_details)  
          return q  
   
       def solve_prec1(self,p):  
      #proj=Projector(domain=self.domain, reduce = True, fast=False)  
          self.__pde_prec.setTolerance(self.getSubProblemTolerance())  
          self.__pde_prec.setValue(Y=p)  
          q=self.__pde_prec.getSolution(verbose=self.show_details)  
      q0=util.interpolate(q,Function(self.domain))  
          print util.inf(q*q0),util.sup(q*q0)  
          q-=(1/self.vol)*util.integrate(q0)  
          print util.inf(q*q0),util.sup(q*q0)  
          return q  
   
       def stoppingcriterium(self,Bv,v,p):  
           n_r=util.sqrt(self.inner(Bv,Bv))  
           n_v=util.sqrt(util.integrate(util.length(util.grad(v))**2))  
           if self.verbose: print "PCG step %s: L2(div(v)) = %s, L2(grad(v))=%s"%(self.iter,n_r,n_v)  
           if self.iter == 0: self.__n_v=n_v;  
           self.__n_v, n_v_old =n_v, self.__n_v  
           self.iter+=1  
           print abs(n_v_old-self.__n_v), n_v, self.getTolerance()  
           if self.iter>1 and n_r <= n_v*self.getTolerance() and abs(n_v_old-self.__n_v) <= n_v * self.getTolerance():  
               if self.verbose: print "PCG terminated after %s steps."%self.iter  
               return True  
           else:  
               return False  
       def stoppingcriterium2(self,norm_r,norm_b,solver='GMRES',TOL=None):  
       if TOL==None:  
              TOL=self.getTolerance()  
           if self.verbose: print "%s step %s: L2(r) = %s, L2(b)*TOL=%s"%(solver,self.iter,norm_r,norm_b*TOL)  
           self.iter+=1  
             
           if norm_r <= norm_b*TOL:  
               if self.verbose: print "%s terminated after %s steps."%(solver,self.iter)  
               return True  
           else:  
               return False  
901    
902    
903         def getDV(self, p, v, tol):
904             """
905             return the value for v for a given p (overwrite)
906    
907             :param p: a pressure
908             :param v: a initial guess for the value v to return.
909             :return: dv given as *Adv=(f-Av-B^*p)*
910             """
911             self.updateStokesEquation(v,p)
912             self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress)
913         self.getSolverOptionsVelocity().setTolerance(tol)
914         self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
915             if self.__stress.isEmpty():
916                self.__pde_u.setValue(X=p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
917             else:
918                self.__pde_u.setValue(X=self.__stress+p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
919             out=self.__pde_u.getSolution()
920             return  out
921    
922         def norm_Bv(self,Bv):
923            """
924            Returns Bv (overwrite).
925    
926            :rtype: equal to the type of p
927            :note: boundary conditions on p should be zero!
928            """
929            return util.sqrt(util.integrate(util.interpolate(Bv,Function(self.domain))**2))
930    
931         def solve_AinvBt(self,p, tol):
932             """
933             Solves *Av=B^*p* with accuracy `tol`
934    
935             :param p: a pressure increment
936             :return: the solution of *Av=B^*p*
937             :note: boundary conditions on v should be zero!
938             """
939             self.__pde_u.setValue(Y=Data(), y=Data(), X=-p*util.kronecker(self.domain))
940             out=self.__pde_u.getSolution()
941             return  out
942    
943         def solve_prec(self,Bv, tol):
944             """
945             applies preconditioner for for *BA^{-1}B^** to *Bv*
946             with accuracy `self.getSubProblemTolerance()`
947    
948             :param Bv: velocity increment
949             :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
950             :note: boundary conditions on p are zero.
951             """
952             self.__pde_prec.setValue(Y=Bv)
953         self.getSolverOptionsPressure().setTolerance(tol)
954         self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
955             out=self.__pde_prec.getSolution()
956             return out

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