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######################################################## |
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# |
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# Copyright (c) 2003-2008 by University of Queensland |
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# Earth Systems Science Computational Center (ESSCC) |
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# http://www.uq.edu.au/esscc |
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# |
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# Primary Business: Queensland, Australia |
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# Licensed under the Open Software License version 3.0 |
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# http://www.opensource.org/licenses/osl-3.0.php |
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# |
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######################################################## |
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|
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__copyright__="""Copyright (c) 2003-2008 by University of Queensland |
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Earth Systems Science Computational Center (ESSCC) |
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http://www.uq.edu.au/esscc |
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Primary Business: Queensland, Australia""" |
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__license__="""Licensed under the Open Software License version 3.0 |
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http://www.opensource.org/licenses/osl-3.0.php""" |
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__url__="http://www.uq.edu.au/esscc/escript-finley" |
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|
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""" |
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Some models for flow |
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|
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@var __author__: name of author |
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@var __copyright__: copyrights |
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@var __license__: licence agreement |
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@var __url__: url entry point on documentation |
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@var __version__: version |
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@var __date__: date of the version |
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""" |
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|
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__author__="Lutz Gross, l.gross@uq.edu.au" |
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|
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from escript import * |
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import util |
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from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE |
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from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES |
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|
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class DarcyFlow(object): |
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""" |
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solves the problem |
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|
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M{u_i+k_{ij}*p_{,j} = g_i} |
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M{u_{i,i} = f} |
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|
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where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability, |
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|
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@note: The problem is solved in a least squares formulation. |
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""" |
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|
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def __init__(self, domain,useReduced=False): |
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""" |
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initializes the Darcy flux problem |
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@param domain: domain of the problem |
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@type domain: L{Domain} |
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""" |
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self.domain=domain |
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self.__l=util.longestEdge(self.domain)**2 |
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self.__pde_v=LinearPDESystem(domain) |
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if useReduced: self.__pde_v.setReducedOrderOn() |
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self.__pde_v.setSymmetryOn() |
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self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain))) |
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self.__pde_p=LinearSinglePDE(domain) |
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self.__pde_p.setSymmetryOn() |
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if useReduced: self.__pde_p.setReducedOrderOn() |
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self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X")) |
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self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y")) |
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self.setTolerance() |
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self.setAbsoluteTolerance() |
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self.setSubProblemTolerance() |
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|
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def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None): |
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""" |
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assigns values to model parameters |
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|
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@param f: volumetic sources/sinks |
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@type f: scalar value on the domain (e.g. L{Data}) |
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@param g: flux sources/sinks |
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@type g: vector values on the domain (e.g. L{Data}) |
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@param location_of_fixed_pressure: mask for locations where pressure is fixed |
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@type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data}) |
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@param location_of_fixed_flux: mask for locations where flux is fixed. |
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@type location_of_fixed_flux: vector values on the domain (e.g. L{Data}) |
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@param permeability: permeability tensor. If scalar C{s} is given the tensor with |
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C{s} on the main diagonal is used. If vector C{v} is given the tensor with |
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C{v} on the main diagonal is used. |
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@type permeability: scalar, vector or tensor values on the domain (e.g. L{Data}) |
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|
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@note: the values of parameters which are not set by calling C{setValue} are not altered. |
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@note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0) |
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or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal |
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is along the M{x_i} axis. |
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""" |
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if f !=None: |
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f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X")) |
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if f.isEmpty(): |
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f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X")) |
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else: |
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if f.getRank()>0: raise ValueError,"illegal rank of f." |
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self.__f=f |
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if g !=None: |
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g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y")) |
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if g.isEmpty(): |
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g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y")) |
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else: |
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if not g.getShape()==(self.domain.getDim(),): |
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raise ValueError,"illegal shape of g" |
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self.__g=g |
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|
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if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure) |
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if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux) |
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|
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if permeability!=None: |
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perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A")) |
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if perm.getRank()==0: |
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perm=perm*util.kronecker(self.domain.getDim()) |
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elif perm.getRank()==1: |
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perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm |
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for i in range(self.domain.getDim()): perm[i,i]=perm2[i] |
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elif perm.getRank()==2: |
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pass |
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else: |
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raise ValueError,"illegal rank of permeability." |
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self.__permeability=perm |
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self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability)) |
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|
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def setTolerance(self,rtol=1e-4): |
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""" |
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sets the relative tolerance C{rtol} used to terminate the solution process. The iteration is terminated if |
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|
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M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) } |
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|
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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}}. |
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|
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@param rtol: relative tolerance for the pressure |
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@type rtol: non-negative C{float} |
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""" |
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if rtol<0: |
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raise ValueError,"Relative tolerance needs to be non-negative." |
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self.__rtol=rtol |
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def getTolerance(self): |
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""" |
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returns the relative tolerance |
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|
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@return: current relative tolerance |
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@rtype: C{float} |
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""" |
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return self.__rtol |
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|
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def setAbsoluteTolerance(self,atol=0.): |
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""" |
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sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if |
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|
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M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) } |
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|
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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}}. |
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|
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@param atol: absolute tolerance for the pressure |
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@type atol: non-negative C{float} |
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""" |
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if atol<0: |
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raise ValueError,"Absolute tolerance needs to be non-negative." |
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self.__atol=atol |
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def getAbsoluteTolerance(self): |
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""" |
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returns the absolute tolerance |
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|
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@return: current absolute tolerance |
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@rtype: C{float} |
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""" |
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return self.__atol |
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|
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def setSubProblemTolerance(self,rtol=None): |
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""" |
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Sets the relative tolerance to solve the subproblem(s). If C{rtol} is not present |
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C{self.getTolerance()**2} is used. |
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|
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@param rtol: relative tolerence |
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@type rtol: positive C{float} |
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""" |
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if rtol == None: |
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if self.getTolerance()<=0.: |
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raise ValueError,"A positive relative tolerance must be set." |
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self.__sub_tol=max(util.EPSILON**(0.75),self.getTolerance()**2) |
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else: |
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if rtol<=0: |
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raise ValueError,"sub-problem tolerance must be positive." |
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self.__sub_tol=max(util.EPSILON**(0.75),rtol) |
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|
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def getSubProblemTolerance(self): |
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""" |
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Returns the subproblem reduction factor. |
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|
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@return: subproblem reduction factor |
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@rtype: C{float} |
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""" |
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return self.__sub_tol |
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|
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def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, max_num_corrections=10): |
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""" |
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solves the problem. |
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|
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The iteration is terminated if the residual norm is less then self.getTolerance(). |
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|
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@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. |
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@type u0: vector value on the domain (e.g. L{Data}). |
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@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. |
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@type p0: scalar value on the domain (e.g. L{Data}). |
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@param verbose: if set some information on iteration progress are printed |
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@type verbose: C{bool} |
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@param show_details: if set information on the subiteration process are printed. |
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@type show_details: C{bool} |
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@return: flux and pressure |
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@rtype: C{tuple} of L{Data}. |
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|
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@note: The problem is solved as a least squares form |
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|
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M{(I+D^*D)u+Qp=D^*f+g} |
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M{Q^*u+Q^*Qp=Q^*g} |
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|
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where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}. |
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We eliminate the flux form the problem by setting |
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|
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M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux |
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|
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form the first equation. Inserted into the second equation we get |
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|
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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 |
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|
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which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step |
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PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme. |
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""" |
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self.verbose=verbose or True |
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self.show_details= show_details and self.verbose |
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rtol=self.getTolerance() |
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atol=self.getAbsoluteTolerance() |
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if self.verbose: print "DarcyFlux: initial sub tolerance = %e"%self.getSubProblemTolerance() |
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|
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num_corrections=0 |
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converged=False |
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p=p0 |
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norm_r=None |
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while not converged: |
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v=self.getFlux(p, fixed_flux=u0, show_details=self.show_details) |
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Qp=self.__Q(p) |
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norm_v=self.__L2(v) |
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norm_Qp=self.__L2(Qp) |
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if norm_v == 0.: |
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if norm_Qp == 0.: |
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return v,p |
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else: |
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fac=norm_Qp |
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else: |
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if norm_Qp == 0.: |
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fac=norm_v |
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else: |
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fac=2./(1./norm_v+1./norm_Qp) |
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ATOL=(atol+rtol*fac) |
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if self.verbose: |
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print "DarcyFlux: L2 norm of v = %e."%norm_v |
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print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp |
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print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL |
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if norm_r == None or norm_r>ATOL: |
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if num_corrections>max_num_corrections: |
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raise ValueError,"maximum number of correction steps reached." |
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p,r, norm_r=PCG(self.__g-util.interpolate(v,Function(self.domain))-Qp,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.1*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose) |
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num_corrections+=1 |
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else: |
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converged=True |
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return v,p |
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# |
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# |
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# r_hat=g-util.interpolate(v,Function(self.domain))-Qp |
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# #=========================================================================== |
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# norm_r_hat=self.__L2(r_hat) |
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# norm_v=self.__L2(v) |
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# norm_g=self.__L2(g) |
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# norm_gv=self.__L2(g-v) |
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# norm_Qp=self.__L2(Qp) |
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# norm_gQp=self.__L2(g-Qp) |
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# fac=min(max(norm_v,norm_gQp),max(norm_Qp,norm_gv)) |
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# fac=min(norm_v,norm_Qp,norm_gv) |
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# norm_r_hat_PCG=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(r_hat),r_hat)) |
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# print "norm_r_hat = ",norm_r_hat,norm_r_hat_PCG, norm_r_hat_PCG/norm_r_hat |
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# if r!=None: |
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# print "diff = ",self.__L2(r-r_hat)/norm_r_hat |
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# sub_tol=min(rtol/self.__L2(r-r_hat)*norm_r_hat,1.)*self.getSubProblemTolerance() |
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# self.setSubProblemTolerance(sub_tol) |
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# print "subtol_new=",self.getSubProblemTolerance() |
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# print "norm_v = ",norm_v |
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# print "norm_gv = ",norm_gv |
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# print "norm_Qp = ",norm_Qp |
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# print "norm_gQp = ",norm_gQp |
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# print "norm_g = ",norm_g |
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# print "max(norm_v,norm_gQp)=",max(norm_v,norm_gQp) |
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# print "max(norm_Qp,norm_gv)=",max(norm_Qp,norm_gv) |
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# if fac == 0: |
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# if self.verbose: print "DarcyFlux: trivial case!" |
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# return v,p |
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# #=============================================================================== |
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# # norm_v=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(v),v)) |
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# # norm_Qp=self.__L2(Qp) |
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# norm_r_hat=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(r_hat),r_hat)) |
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# # print "**** norm_v, norm_Qp :",norm_v,norm_Qp |
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# |
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# ATOL=(atol+rtol*2./(1./norm_v+1./norm_Qp)) |
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# if self.verbose: |
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# print "DarcyFlux: residual = %e"%norm_r_hat |
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# print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL |
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# if norm_r_hat <= ATOL: |
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# print "DarcyFlux: iteration finalized." |
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# converged=True |
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# else: |
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# # p=GMRES(r_hat,self.__Aprod, p, self.__inner_GMRES, atol=ATOL, rtol=0., iter_max=max_iter, iter_restart=20, verbose=self.verbose,P_R=self.__Msolve_PCG) |
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# # p,r=PCG(r_hat,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL*min(0.1,norm_r_hat_PCG/norm_r_hat), rtol=0.,iter_max=max_iter, verbose=self.verbose) |
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# p,r, norm_r=PCG(r_hat,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.1*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose) |
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# print "norm_r =",norm_r |
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# return v,p |
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def __L2(self,v): |
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return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2)) |
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|
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def __Q(self,p): |
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return util.tensor_mult(self.__permeability,util.grad(p)) |
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|
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def __Aprod(self,dp): |
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self.__pde_v.setTolerance(self.getSubProblemTolerance()) |
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if self.show_details: print "DarcyFlux: Applying operator" |
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Qdp=self.__Q(dp) |
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self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data()) |
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du=self.__pde_v.getSolution(verbose=self.show_details) |
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return Qdp+du |
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def __inner_GMRES(self,r,s): |
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return util.integrate(util.inner(r,s)) |
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|
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def __inner_PCG(self,p,r): |
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return util.integrate(util.inner(self.__Q(p), r)) |
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|
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def __Msolve_PCG(self,r): |
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self.__pde_p.setTolerance(self.getSubProblemTolerance()) |
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if self.show_details: print "DarcyFlux: Applying preconditioner" |
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self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data()) |
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return self.__pde_p.getSolution(verbose=self.show_details) |
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|
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def getFlux(self,p=None, fixed_flux=Data(), show_details=False): |
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""" |
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returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux} |
| 347 |
on locations where C{location_of_fixed_flux} is positive (see L{setValue}). |
| 348 |
Note that C{g} and C{f} are used, see L{setValue}. |
| 349 |
|
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@param p: pressure. |
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@type p: scalar value on the domain (e.g. L{Data}). |
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@param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}. |
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@type fixed_flux: vector values on the domain (e.g. L{Data}). |
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@param tol: relative tolerance to be used. |
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@type tol: positive C{float}. |
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@return: flux |
| 357 |
@rtype: L{Data} |
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@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}} |
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for the permeability M{k_{ij}} |
| 360 |
""" |
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self.__pde_v.setTolerance(self.getSubProblemTolerance()) |
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g=self.__g |
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f=self.__f |
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self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux) |
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if p == None: |
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self.__pde_v.setValue(Y=g) |
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else: |
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self.__pde_v.setValue(Y=g-self.__Q(p)) |
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return self.__pde_v.getSolution(verbose=show_details) |
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|
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class StokesProblemCartesian(HomogeneousSaddlePointProblem): |
| 372 |
""" |
| 373 |
solves |
| 374 |
|
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-(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j} |
| 376 |
u_{i,i}=0 |
| 377 |
|
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u=0 where fixed_u_mask>0 |
| 379 |
eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j |
| 380 |
|
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if surface_stress is not given 0 is assumed. |
| 382 |
|
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typical usage: |
| 384 |
|
| 385 |
sp=StokesProblemCartesian(domain) |
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sp.setTolerance() |
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sp.initialize(...) |
| 388 |
v,p=sp.solve(v0,p0) |
| 389 |
""" |
| 390 |
def __init__(self,domain,**kwargs): |
| 391 |
""" |
| 392 |
initialize the Stokes Problem |
| 393 |
|
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@param domain: domain of the problem. The approximation order needs to be two. |
| 395 |
@type domain: L{Domain} |
| 396 |
@warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure. |
| 397 |
""" |
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HomogeneousSaddlePointProblem.__init__(self,**kwargs) |
| 399 |
self.domain=domain |
| 400 |
self.vol=util.integrate(1.,Function(self.domain)) |
| 401 |
self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim()) |
| 402 |
self.__pde_u.setSymmetryOn() |
| 403 |
# self.__pde_u.setSolverMethod(self.__pde_u.DIRECT) |
| 404 |
# self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU) |
| 405 |
|
| 406 |
self.__pde_prec=LinearPDE(domain) |
| 407 |
self.__pde_prec.setReducedOrderOn() |
| 408 |
# self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING) |
| 409 |
self.__pde_prec.setSymmetryOn() |
| 410 |
|
| 411 |
def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()): |
| 412 |
""" |
| 413 |
assigns values to the model parameters |
| 414 |
|
| 415 |
@param f: external force |
| 416 |
@type f: L{Vector} object in L{FunctionSpace} L{Function} or similar |
| 417 |
@param fixed_u_mask: mask of locations with fixed velocity. |
| 418 |
@type fixed_u_mask: L{Vector} object on L{FunctionSpace} L{Solution} or similar |
| 419 |
@param eta: viscosity |
| 420 |
@type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar |
| 421 |
@param surface_stress: normal surface stress |
| 422 |
@type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar |
| 423 |
@param stress: initial stress |
| 424 |
@type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar |
| 425 |
@note: All values needs to be set. |
| 426 |
|
| 427 |
""" |
| 428 |
self.eta=eta |
| 429 |
A =self.__pde_u.createCoefficient("A") |
| 430 |
self.__pde_u.setValue(A=Data()) |
| 431 |
for i in range(self.domain.getDim()): |
| 432 |
for j in range(self.domain.getDim()): |
| 433 |
A[i,j,j,i] += 1. |
| 434 |
A[i,j,i,j] += 1. |
| 435 |
self.__pde_prec.setValue(D=1/self.eta) |
| 436 |
self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask) |
| 437 |
self.__f=f |
| 438 |
self.__surface_stress=surface_stress |
| 439 |
self.__stress=stress |
| 440 |
|
| 441 |
def inner_pBv(self,p,v): |
| 442 |
""" |
| 443 |
returns inner product of element p and div(v) |
| 444 |
|
| 445 |
@param p: a pressure increment |
| 446 |
@param v: a residual |
| 447 |
@return: inner product of element p and div(v) |
| 448 |
@rtype: C{float} |
| 449 |
""" |
| 450 |
return util.integrate(-p*util.div(v)) |
| 451 |
|
| 452 |
def inner_p(self,p0,p1): |
| 453 |
""" |
| 454 |
Returns inner product of p0 and p1 |
| 455 |
|
| 456 |
@param p0: a pressure |
| 457 |
@param p1: a pressure |
| 458 |
@return: inner product of p0 and p1 |
| 459 |
@rtype: C{float} |
| 460 |
""" |
| 461 |
s0=util.interpolate(p0/self.eta,Function(self.domain)) |
| 462 |
s1=util.interpolate(p1/self.eta,Function(self.domain)) |
| 463 |
return util.integrate(s0*s1) |
| 464 |
|
| 465 |
def norm_v(self,v): |
| 466 |
""" |
| 467 |
returns the norm of v |
| 468 |
|
| 469 |
@param v: a velovity |
| 470 |
@return: norm of v |
| 471 |
@rtype: non-negative C{float} |
| 472 |
""" |
| 473 |
return util.sqrt(util.integrate(util.length(util.grad(v)))) |
| 474 |
|
| 475 |
def getV(self, p, v0): |
| 476 |
""" |
| 477 |
return the value for v for a given p (overwrite) |
| 478 |
|
| 479 |
@param p: a pressure |
| 480 |
@param v0: a initial guess for the value v to return. |
| 481 |
@return: v given as M{v= A^{-1} (f-B^*p)} |
| 482 |
""" |
| 483 |
self.__pde_u.setTolerance(self.getSubProblemTolerance()) |
| 484 |
self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress, r=v0) |
| 485 |
if self.__stress.isEmpty(): |
| 486 |
self.__pde_u.setValue(X=p*util.kronecker(self.domain)) |
| 487 |
else: |
| 488 |
self.__pde_u.setValue(X=self.__stress+p*util.kronecker(self.domain)) |
| 489 |
out=self.__pde_u.getSolution(verbose=self.show_details) |
| 490 |
return out |
| 491 |
|
| 492 |
|
| 493 |
raise NotImplementedError,"no v calculation implemented." |
| 494 |
|
| 495 |
|
| 496 |
def norm_Bv(self,v): |
| 497 |
""" |
| 498 |
Returns Bv (overwrite). |
| 499 |
|
| 500 |
@rtype: equal to the type of p |
| 501 |
@note: boundary conditions on p should be zero! |
| 502 |
""" |
| 503 |
return util.sqrt(util.integrate(util.div(v)**2)) |
| 504 |
|
| 505 |
def solve_AinvBt(self,p): |
| 506 |
""" |
| 507 |
Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()} |
| 508 |
|
| 509 |
@param p: a pressure increment |
| 510 |
@return: the solution of M{Av=B^*p} |
| 511 |
@note: boundary conditions on v should be zero! |
| 512 |
""" |
| 513 |
self.__pde_u.setTolerance(self.getSubProblemTolerance()) |
| 514 |
self.__pde_u.setValue(Y=Data(), y=Data(), r=Data(),X=-p*util.kronecker(self.domain)) |
| 515 |
out=self.__pde_u.getSolution(verbose=self.show_details) |
| 516 |
return out |
| 517 |
|
| 518 |
def solve_precB(self,v): |
| 519 |
""" |
| 520 |
applies preconditioner for for M{BA^{-1}B^*} to M{Bv} |
| 521 |
with accuracy L{self.getSubProblemTolerance()} (overwrite). |
| 522 |
|
| 523 |
@param v: velocity increment |
| 524 |
@return: M{p=P(Bv)} where M{P^{-1}} is an approximation of M{BA^{-1}B^*} |
| 525 |
@note: boundary conditions on p are zero. |
| 526 |
""" |
| 527 |
self.__pde_prec.setValue(Y=-util.div(v)) |
| 528 |
self.__pde_prec.setTolerance(self.getSubProblemTolerance()) |
| 529 |
return self.__pde_prec.getSolution(verbose=self.show_details) |
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