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Revision 2620 - (show annotations)
Thu Aug 20 06:24:00 2009 UTC (10 years, 1 month ago) by gross
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some small additions to pycad to make life a bit easier.
1 ########################################################
2 #
3 # Copyright (c) 2003-2009 by University of Queensland
4 # Earth Systems Science Computational Center (ESSCC)
5 # http://www.uq.edu.au/esscc
6 #
7 # Primary Business: Queensland, Australia
8 # Licensed under the Open Software License version 3.0
9 # http://www.opensource.org/licenses/osl-3.0.php
10 #
11 ########################################################
12
13 __copyright__="""Copyright (c) 2003-2009 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
23
24 @var __author__: name of author
25 @var __copyright__: copyrights
26 @var __license__: licence agreement
27 @var __url__: url entry point on documentation
28 @var __version__: version
29 @var __date__: date of the version
30 """
31
32 __author__="Lutz Gross, l.gross@uq.edu.au"
33
34 from escript import *
35 import util
36 from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
37 from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
38
39 class DarcyFlow(object):
40 """
41 solves the problem
42
43 M{u_i+k_{ij}*p_{,j} = g_i}
44 M{u_{i,i} = f}
45
46 where M{p} represents the pressure and M{u} the Darcy flux. M{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: L{Domain}
56 @param useReduced: uses reduced oreder on flux and pressure
57 @type useReduced: C{bool}
58 @param adaptSubTolerance: switches on automatic subtolerance selection
59 @type adaptSubTolerance: C{bool}
60 """
61 self.domain=domain
62 if weight == None:
63 s=self.domain.getSize()
64 self.__l=(3.*util.longestEdge(self.domain)*s/util.sup(s))**2
65 # self.__l=(3.*util.longestEdge(self.domain))**2
66 # self.__l=(0.1*util.longestEdge(self.domain)*s/util.sup(s))**2
67 else:
68 self.__l=weight
69 self.__pde_v=LinearPDESystem(domain)
70 if useReduced: self.__pde_v.setReducedOrderOn()
71 self.__pde_v.setSymmetryOn()
72 self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain)))
73 self.__pde_p=LinearSinglePDE(domain)
74 self.__pde_p.setSymmetryOn()
75 if useReduced: self.__pde_p.setReducedOrderOn()
76 self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
77 self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
78 self.setTolerance()
79 self.setAbsoluteTolerance()
80 self.__adaptSubTolerance=adaptSubTolerance
81 self.verbose=False
82 def getSolverOptionsFlux(self):
83 """
84 Returns the solver options used to solve the flux problems
85
86 M{(I+D^*D)u=F}
87
88 @return: L{SolverOptions}
89 """
90 return self.__pde_v.getSolverOptions()
91 def setSolverOptionsFlux(self, options=None):
92 """
93 Sets the solver options used to solve the flux problems
94
95 M{(I+D^*D)u=F}
96
97 If C{options} is not present, the options are reset to default
98 @param options: L{SolverOptions}
99 @note: if the adaption of subtolerance is choosen, the tolerance set by C{options} will be overwritten before the solver is called.
100 """
101 return self.__pde_v.setSolverOptions(options)
102 def getSolverOptionsPressure(self):
103 """
104 Returns the solver options used to solve the pressure problems
105
106 M{(Q^*Q)p=Q^*G}
107
108 @return: L{SolverOptions}
109 """
110 return self.__pde_p.getSolverOptions()
111 def setSolverOptionsPressure(self, options=None):
112 """
113 Sets the solver options used to solve the pressure problems
114
115 M{(Q^*Q)p=Q^*G}
116
117 If C{options} is not present, the options are reset to default
118 @param options: L{SolverOptions}
119 @note: if the adaption of subtolerance is choosen, the tolerance set by C{options} will be overwritten before the solver is called.
120 """
121 return self.__pde_p.setSolverOptions(options)
122
123 def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
124 """
125 assigns values to model parameters
126
127 @param f: volumetic sources/sinks
128 @type f: scalar value on the domain (e.g. L{Data})
129 @param g: flux sources/sinks
130 @type g: vector values on the domain (e.g. L{Data})
131 @param location_of_fixed_pressure: mask for locations where pressure is fixed
132 @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})
133 @param location_of_fixed_flux: mask for locations where flux is fixed.
134 @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})
135 @param permeability: permeability tensor. If scalar C{s} is given the tensor with
136 C{s} on the main diagonal is used. If vector C{v} is given the tensor with
137 C{v} on the main diagonal is used.
138 @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})
139
140 @note: the values of parameters which are not set by calling C{setValue} are not altered.
141 @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)
142 or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal
143 is along the M{x_i} axis.
144 """
145 if f !=None:
146 f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
147 if f.isEmpty():
148 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
149 else:
150 if f.getRank()>0: raise ValueError,"illegal rank of f."
151 self.__f=f
152 if g !=None:
153 g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
154 if g.isEmpty():
155 g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
156 else:
157 if not g.getShape()==(self.domain.getDim(),):
158 raise ValueError,"illegal shape of g"
159 self.__g=g
160
161 if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)
162 if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)
163
164 if permeability!=None:
165 perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
166 if perm.getRank()==0:
167 perm=perm*util.kronecker(self.domain.getDim())
168 elif perm.getRank()==1:
169 perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm
170 for i in range(self.domain.getDim()): perm[i,i]=perm2[i]
171 elif perm.getRank()==2:
172 pass
173 else:
174 raise ValueError,"illegal rank of permeability."
175 self.__permeability=perm
176 self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))
177
178 def setTolerance(self,rtol=1e-4):
179 """
180 sets the relative tolerance C{rtol} used to terminate the solution process. The iteration is terminated if
181
182 M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
183
184 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}}.
185
186 @param rtol: relative tolerance for the pressure
187 @type rtol: non-negative C{float}
188 """
189 if rtol<0:
190 raise ValueError,"Relative tolerance needs to be non-negative."
191 self.__rtol=rtol
192 def getTolerance(self):
193 """
194 returns the relative tolerance
195
196 @return: current relative tolerance
197 @rtype: C{float}
198 """
199 return self.__rtol
200
201 def setAbsoluteTolerance(self,atol=0.):
202 """
203 sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if
204
205 M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
206
207 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}}.
208
209 @param atol: absolute tolerance for the pressure
210 @type atol: non-negative C{float}
211 """
212 if atol<0:
213 raise ValueError,"Absolute tolerance needs to be non-negative."
214 self.__atol=atol
215 def getAbsoluteTolerance(self):
216 """
217 returns the absolute tolerance
218
219 @return: current absolute tolerance
220 @rtype: C{float}
221 """
222 return self.__atol
223 def getSubProblemTolerance(self):
224 """
225 Returns a suitable subtolerance
226 @type: C{float}
227 """
228 return max(util.EPSILON**(0.75),self.getTolerance()**2)
229 def setSubProblemTolerance(self):
230 """
231 Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected.
232 """
233 if self.__adaptSubTolerance:
234 sub_tol=self.getSubProblemTolerance()
235 self.getSolverOptionsFlux().setTolerance(sub_tol)
236 self.getSolverOptionsFlux().setAbsoluteTolerance(0.)
237 self.getSolverOptionsPressure().setTolerance(sub_tol)
238 self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
239 if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol
240
241 def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10):
242 """
243 solves the problem.
244
245 The iteration is terminated if the residual norm is less then self.getTolerance().
246
247 @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.
248 @type u0: vector value on the domain (e.g. L{Data}).
249 @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.
250 @type p0: scalar value on the domain (e.g. L{Data}).
251 @param verbose: if set some information on iteration progress are printed
252 @type verbose: C{bool}
253 @return: flux and pressure
254 @rtype: C{tuple} of L{Data}.
255
256 @note: The problem is solved as a least squares form
257
258 M{(I+D^*D)u+Qp=D^*f+g}
259 M{Q^*u+Q^*Qp=Q^*g}
260
261 where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
262 We eliminate the flux form the problem by setting
263
264 M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux
265
266 form the first equation. Inserted into the second equation we get
267
268 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
269
270 which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step
271 PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.
272 """
273 self.verbose=verbose
274 rtol=self.getTolerance()
275 atol=self.getAbsoluteTolerance()
276 self.setSubProblemTolerance()
277
278 num_corrections=0
279 converged=False
280 p=p0
281 norm_r=None
282 while not converged:
283 v=self.getFlux(p, fixed_flux=u0)
284 Qp=self.__Q(p)
285 norm_v=self.__L2(v)
286 norm_Qp=self.__L2(Qp)
287 if norm_v == 0.:
288 if norm_Qp == 0.:
289 return v,p
290 else:
291 fac=norm_Qp
292 else:
293 if norm_Qp == 0.:
294 fac=norm_v
295 else:
296 fac=2./(1./norm_v+1./norm_Qp)
297 ATOL=(atol+rtol*fac)
298 if self.verbose:
299 print "DarcyFlux: L2 norm of v = %e."%norm_v
300 print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp
301 print "DarcyFlux: L2 defect u = %e."%(util.integrate(util.length(self.__g-util.interpolate(v,Function(self.domain))-Qp)**2)**(0.5),)
302 print "DarcyFlux: L2 defect div(v) = %e."%(util.integrate((self.__f-util.div(v))**2)**(0.5),)
303 print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
304 if norm_r == None or norm_r>ATOL:
305 if num_corrections>max_num_corrections:
306 raise ValueError,"maximum number of correction steps reached."
307 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)
308 num_corrections+=1
309 else:
310 converged=True
311 return v,p
312 def __L2(self,v):
313 return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
314
315 def __Q(self,p):
316 return util.tensor_mult(self.__permeability,util.grad(p))
317
318 def __Aprod(self,dp):
319 if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator"
320 Qdp=self.__Q(dp)
321 self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())
322 du=self.__pde_v.getSolution()
323 # self.__pde_v.getOperator().saveMM("proj.mm")
324 return Qdp+du
325 def __inner_GMRES(self,r,s):
326 return util.integrate(util.inner(r,s))
327
328 def __inner_PCG(self,p,r):
329 return util.integrate(util.inner(self.__Q(p), r))
330
331 def __Msolve_PCG(self,r):
332 if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner"
333 self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data())
334 # self.__pde_p.getOperator().saveMM("prec.mm")
335 return self.__pde_p.getSolution()
336
337 def getFlux(self,p=None, fixed_flux=Data()):
338 """
339 returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}
340 on locations where C{location_of_fixed_flux} is positive (see L{setValue}).
341 Note that C{g} and C{f} are used, see L{setValue}.
342
343 @param p: pressure.
344 @type p: scalar value on the domain (e.g. L{Data}).
345 @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.
346 @type fixed_flux: vector values on the domain (e.g. L{Data}).
347 @param tol: relative tolerance to be used.
348 @type tol: positive C{float}.
349 @return: flux
350 @rtype: L{Data}
351 @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}}
352 for the permeability M{k_{ij}}
353 """
354 self.setSubProblemTolerance()
355 g=self.__g
356 f=self.__f
357 self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)
358 if p == None:
359 self.__pde_v.setValue(Y=g)
360 else:
361 self.__pde_v.setValue(Y=g-self.__Q(p))
362 return self.__pde_v.getSolution()
363
364 class StokesProblemCartesian(HomogeneousSaddlePointProblem):
365 """
366 solves
367
368 -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
369 u_{i,i}=0
370
371 u=0 where fixed_u_mask>0
372 eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
373
374 if surface_stress is not given 0 is assumed.
375
376 typical usage:
377
378 sp=StokesProblemCartesian(domain)
379 sp.setTolerance()
380 sp.initialize(...)
381 v,p=sp.solve(v0,p0)
382 """
383 def __init__(self,domain,adaptSubTolerance=True, **kwargs):
384 """
385 initialize the Stokes Problem
386
387 @param domain: domain of the problem. The approximation order needs to be two.
388 @type domain: L{Domain}
389 @param adaptSubTolerance: If True the tolerance for subproblem is set automatically.
390 @type adaptSubTolerance: C{bool}
391 @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.
392 """
393 HomogeneousSaddlePointProblem.__init__(self,adaptSubTolerance=adaptSubTolerance,**kwargs)
394 self.domain=domain
395 self.vol=util.integrate(1.,Function(self.domain))
396 self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
397 self.__pde_u.setSymmetryOn()
398
399 self.__pde_prec=LinearPDE(domain)
400 self.__pde_prec.setReducedOrderOn()
401 self.__pde_prec.setSymmetryOn()
402
403 self.__pde_proj=LinearPDE(domain)
404 self.__pde_proj.setReducedOrderOn()
405 self.__pde_proj.setValue(D=1)
406 self.__pde_proj.setSymmetryOn()
407
408 def getSolverOptionsVelocity(self):
409 """
410 returns the solver options used solve the equation for velocity.
411
412 @rtype: L{SolverOptions}
413 """
414 return self.__pde_u.getSolverOptions()
415 def setSolverOptionsVelocity(self, options=None):
416 """
417 set the solver options for solving the equation for velocity.
418
419 @param options: new solver options
420 @type options: L{SolverOptions}
421 """
422 self.__pde_u.setSolverOptions(options)
423 def getSolverOptionsPressure(self):
424 """
425 returns the solver options used solve the equation for pressure.
426 @rtype: L{SolverOptions}
427 """
428 return self.__pde_prec.getSolverOptions()
429 def setSolverOptionsPressure(self, options=None):
430 """
431 set the solver options for solving the equation for pressure.
432 @param options: new solver options
433 @type options: L{SolverOptions}
434 """
435 self.__pde_prec.setSolverOptions(options)
436
437 def setSolverOptionsDiv(self, options=None):
438 """
439 set the solver options for solving the equation to project the divergence of
440 the velocity onto the function space of presure.
441
442 @param options: new solver options
443 @type options: L{SolverOptions}
444 """
445 self.__pde_prec.setSolverOptions(options)
446 def getSolverOptionsDiv(self):
447 """
448 returns the solver options for solving the equation to project the divergence of
449 the velocity onto the function space of presure.
450
451 @rtype: L{SolverOptions}
452 """
453 return self.__pde_prec.getSolverOptions()
454 def setSubProblemTolerance(self):
455 """
456 Updates the tolerance for subproblems
457 """
458 if self.adaptSubTolerance():
459 sub_tol=self.getSubProblemTolerance()
460 self.getSolverOptionsDiv().setTolerance(sub_tol)
461 self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
462 self.getSolverOptionsPressure().setTolerance(sub_tol)
463 self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
464 self.getSolverOptionsVelocity().setTolerance(sub_tol)
465 self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
466
467
468 def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data(), restoration_factor=0):
469 """
470 assigns values to the model parameters
471
472 @param f: external force
473 @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar
474 @param fixed_u_mask: mask of locations with fixed velocity.
475 @type fixed_u_mask: L{Vector} object on L{FunctionSpace} L{Solution} or similar
476 @param eta: viscosity
477 @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar
478 @param surface_stress: normal surface stress
479 @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar
480 @param stress: initial stress
481 @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar
482 @note: All values needs to be set.
483 """
484 self.eta=eta
485 A =self.__pde_u.createCoefficient("A")
486 self.__pde_u.setValue(A=Data())
487 for i in range(self.domain.getDim()):
488 for j in range(self.domain.getDim()):
489 A[i,j,j,i] += 1.
490 A[i,j,i,j] += 1.
491 n=self.domain.getNormal()
492 self.__pde_prec.setValue(D=1/self.eta)
493 self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask, d=restoration_factor*util.outer(n,n))
494 self.__f=f
495 self.__surface_stress=surface_stress
496 self.__stress=stress
497
498 def Bv(self,v):
499 """
500 returns inner product of element p and div(v)
501
502 @param p: a pressure increment
503 @param v: a residual
504 @return: inner product of element p and div(v)
505 @rtype: C{float}
506 """
507 self.__pde_proj.setValue(Y=-util.div(v))
508 return self.__pde_proj.getSolution()
509
510 def inner_pBv(self,p,Bv):
511 """
512 returns inner product of element p and Bv=-div(v)
513
514 @param p: a pressure increment
515 @param v: a residual
516 @return: inner product of element p and Bv=-div(v)
517 @rtype: C{float}
518 """
519 return util.integrate(util.interpolate(p,Function(self.domain))*util.interpolate(Bv,Function(self.domain)))
520
521 def inner_p(self,p0,p1):
522 """
523 Returns inner product of p0 and p1
524
525 @param p0: a pressure
526 @param p1: a pressure
527 @return: inner product of p0 and p1
528 @rtype: C{float}
529 """
530 s0=util.interpolate(p0/self.eta,Function(self.domain))
531 s1=util.interpolate(p1/self.eta,Function(self.domain))
532 return util.integrate(s0*s1)
533
534 def norm_v(self,v):
535 """
536 returns the norm of v
537
538 @param v: a velovity
539 @return: norm of v
540 @rtype: non-negative C{float}
541 """
542 return util.sqrt(util.integrate(util.length(util.grad(v))))
543
544 def getV(self, p, v0):
545 """
546 return the value for v for a given p (overwrite)
547
548 @param p: a pressure
549 @param v0: a initial guess for the value v to return.
550 @return: v given as M{v= A^{-1} (f-B^*p)}
551 """
552 self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress, r=v0)
553 if self.__stress.isEmpty():
554 self.__pde_u.setValue(X=p*util.kronecker(self.domain))
555 else:
556 self.__pde_u.setValue(X=self.__stress+p*util.kronecker(self.domain))
557 out=self.__pde_u.getSolution()
558 return out
559
560 def norm_Bv(self,Bv):
561 """
562 Returns Bv (overwrite).
563
564 @rtype: equal to the type of p
565 @note: boundary conditions on p should be zero!
566 """
567 return util.sqrt(util.integrate(util.interpolate(Bv,Function(self.domain))**2))
568
569 def solve_AinvBt(self,p):
570 """
571 Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}
572
573 @param p: a pressure increment
574 @return: the solution of M{Av=B^*p}
575 @note: boundary conditions on v should be zero!
576 """
577 self.__pde_u.setValue(Y=Data(), y=Data(), r=Data(),X=-p*util.kronecker(self.domain))
578 out=self.__pde_u.getSolution()
579 return out
580
581 def solve_prec(self,Bv):
582 """
583 applies preconditioner for for M{BA^{-1}B^*} to M{Bv}
584 with accuracy L{self.getSubProblemTolerance()}
585
586 @param v: velocity increment
587 @return: M{p=P(Bv)} where M{P^{-1}} is an approximation of M{BA^{-1}B^*}
588 @note: boundary conditions on p are zero.
589 """
590 self.__pde_prec.setValue(Y=Bv)
591 return self.__pde_prec.getSolution()

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