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Contents of /trunk/escriptcore/py_src/flows.py

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1 # -*- coding: utf-8 -*-
2 ##############################################################################
3 #
4 # Copyright (c) 2003-2013 by University of Queensland
5 # http://www.uq.edu.au
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 # Development until 2012 by Earth Systems Science Computational Center (ESSCC)
12 # Development since 2012 by School of Earth Sciences
13 #
14 ##############################################################################
15
16 __copyright__="""Copyright (c) 2003-2013 by University of Queensland
17 http://www.uq.edu.au
18 Primary Business: Queensland, Australia"""
19 __license__="""Licensed under the Open Software License version 3.0
20 http://www.opensource.org/licenses/osl-3.0.php"""
21 __url__="https://launchpad.net/escript-finley"
22
23 """
24 Some models for flow
25
26 :var __author__: name of author
27 :var __copyright__: copyrights
28 :var __license__: licence agreement
29 :var __url__: url entry point on documentation
30 :var __version__: version
31 :var __date__: date of the version
32 """
33
34 __author__="Lutz Gross, l.gross@uq.edu.au"
35
36 from . import escriptcpp as escore
37 from . import util
38 from . import linearPDEs as lpe
39 from . import pdetools as pdt
40
41 class DarcyFlow(object):
42 """
43 solves the problem
44
45 *u_i+k_{ij}*p_{,j} = g_i*
46 *u_{i,i} = f*
47
48 where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
49
50 :cvar EVAL: direct pressure gradient evaluation for flux
51 :cvar POST: global postprocessing of flux by solving the PDE *K_{ij} u_j + (w * K * l u_{k,k})_{,i}= - p_{,j} + K_{ij} g_j*
52 where *l* is the length scale, *K* is the inverse of the permeability tensor, and *w* is a positive weighting factor.
53 :cvar SMOOTH: global smoothing by solving the PDE *K_{ij} u_j= - p_{,j} + K_{ij} g_j*
54 """
55 EVAL="EVAL"
56 SIMPLE="EVAL"
57 POST="POST"
58 SMOOTH="SMOOTH"
59 def __init__(self, domain, useReduced=False, solver="POST", verbose=False, w=1.):
60 """
61 initializes the Darcy flux problem.
62
63 :param domain: domain of the problem
64 :type domain: `Domain`
65 :param useReduced: uses reduced oreder on flux and pressure
66 :type useReduced: ``bool``
67 :param solver: solver method
68 :type solver: in [`DarcyFlow.EVAL`, `DarcyFlow.POST`, `DarcyFlow.SMOOTH` ]
69 :param verbose: if ``True`` some information on the iteration progress are printed.
70 :type verbose: ``bool``
71 :param w: weighting factor for `DarcyFlow.POST` solver
72 :type w: ``float``
73
74 """
75 if not solver in [DarcyFlow.EVAL, DarcyFlow.POST, DarcyFlow.SMOOTH ] :
76 raise ValueError("unknown solver %d."%solver)
77
78 self.domain=domain
79 self.solver=solver
80 self.useReduced=useReduced
81 self.verbose=verbose
82 self.l=None
83 self.w=None
84
85 self.__pde_p=lpe.LinearSinglePDE(domain)
86 self.__pde_p.setSymmetryOn()
87 if self.useReduced: self.__pde_p.setReducedOrderOn()
88
89 if self.solver == self.EVAL:
90 self.__pde_v=None
91 if self.verbose: print("DarcyFlow: simple solver is used.")
92
93 elif self.solver == self.POST:
94 if util.inf(w)<0.:
95 raise ValueError("Weighting factor must be non-negative.")
96 if self.verbose: print("DarcyFlow: global postprocessing of flux is used.")
97 self.__pde_v=lpe.LinearPDESystem(domain)
98 self.__pde_v.setSymmetryOn()
99 if self.useReduced: self.__pde_v.setReducedOrderOn()
100 self.w=w
101 x=self.domain.getX()
102 self.l=min( [util.sup(x[i])-util.inf(x[i]) for i in range(self.domain.getDim()) ] )
103 #self.l=util.vol(self.domain)**(1./self.domain.getDim()) # length scale
104
105 elif self.solver == self.SMOOTH:
106 self.__pde_v=lpe.LinearPDESystem(domain)
107 self.__pde_v.setSymmetryOn()
108 if self.useReduced: self.__pde_v.setReducedOrderOn()
109 if self.verbose: print("DarcyFlow: flux smoothing is used.")
110 self.w=0
111
112 self.__f=escore.Data(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
113 self.__g=escore.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
114 self.__permeability_invXg=escore.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
115 self.__permeability_invXg_ref=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
116 self.ref_point_id=None
117 self.ref_point=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
118 self.location_of_fixed_pressure = escore.Data(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
119 self.location_of_fixed_flux = escore.Vector(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
120 self.perm_scale=1.
121
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. `escript.Data`)
129 :param g: flux sources/sinks
130 :type g: vector values on the domain (e.g. `escript.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. `escript.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. `escript.Data`)
135 :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
136 :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
137
138 :note: the values of parameters which are not set by calling ``setValue`` are not altered.
139 :note: at any point on the boundary of the domain the pressure
140 (``location_of_fixed_pressure`` >0) or the normal component of the
141 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
142 is along the *x_i* axis.
143
144 """
145 if location_of_fixed_pressure!=None:
146 self.location_of_fixed_pressure=util.wherePositive(util.interpolate(location_of_fixed_pressure, self.__pde_p.getFunctionSpaceForCoefficient("q")))
147 self.ref_point_id=self.location_of_fixed_pressure.maxGlobalDataPoint()
148 if not self.location_of_fixed_pressure.getTupleForGlobalDataPoint(*self.ref_point_id)[0] > 0: raise ValueError("pressure needs to be fixed at least one point.")
149 self.ref_point=self.__pde_p.getFunctionSpaceForCoefficient("q").getX().getTupleForGlobalDataPoint(*self.ref_point_id)
150 if self.verbose: print(("DarcyFlow: reference point at %s."%(self.ref_point,)))
151 self.__pde_p.setValue(q=self.location_of_fixed_pressure)
152 if location_of_fixed_flux!=None:
153 self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
154 if not self.__pde_v == None:
155 self.__pde_v.setValue(q=self.location_of_fixed_flux)
156
157 if permeability!=None:
158 perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
159 self.perm_scale=util.Lsup(util.length(perm))
160 if self.verbose: print(("DarcyFlow: permeability scaling factor = %e."%self.perm_scale))
161 perm=perm*(1./self.perm_scale)
162
163 if perm.getRank()==0:
164
165 perm_inv=(1./perm)
166 perm_inv=perm_inv*util.kronecker(self.domain.getDim())
167 perm=perm*util.kronecker(self.domain.getDim())
168 elif perm.getRank()==2:
169 perm_inv=util.inverse(perm)
170 else:
171 raise ValueError("illegal rank of permeability.")
172
173 self.__permeability=perm
174 self.__permeability_inv=perm_inv
175
176 #====================
177 self.__pde_p.setValue(A=self.__permeability)
178 if self.solver == self.EVAL:
179 pass # no extra work required
180 elif self.solver == self.POST:
181 k=util.kronecker(self.domain.getDim())
182 self.omega = self.w*util.length(perm_inv)*self.l*self.domain.getSize()
183 #self.__pde_v.setValue(D=self.__permeability_inv, A=self.omega*util.outer(k,k))
184 self.__pde_v.setValue(D=self.__permeability_inv, A_reduced=self.omega*util.outer(k,k))
185 elif self.solver == self.SMOOTH:
186 self.__pde_v.setValue(D=self.__permeability_inv)
187
188 if g != None:
189 g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
190 if g.isEmpty():
191 g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
192 else:
193 if not g.getShape()==(self.domain.getDim(),): raise ValueError("illegal shape of g")
194 self.__g=g
195 self.__permeability_invXg=util.tensor_mult(self.__permeability_inv,self.__g * (1./self.perm_scale ))
196 self.__permeability_invXg_ref=util.integrate(self.__permeability_invXg)/util.vol(self.domain)
197 if f !=None:
198 f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
199 if f.isEmpty():
200 f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
201 else:
202 if f.getRank()>0: raise ValueError("illegal rank of f.")
203 self.__f=f
204
205 def getSolverOptionsFlux(self):
206 """
207 Returns the solver options used to solve the flux problems
208 :return: `SolverOptions`
209 """
210 if self.__pde_v == None:
211 return None
212 else:
213 return self.__pde_v.getSolverOptions()
214
215 def setSolverOptionsFlux(self, options=None):
216 """
217 Sets the solver options used to solve the flux problems
218 If ``options`` is not present, the options are reset to default
219 :param options: `SolverOptions`
220 """
221 if not self.__pde_v == None:
222 self.__pde_v.setSolverOptions(options)
223
224 def getSolverOptionsPressure(self):
225 """
226 Returns the solver options used to solve the pressure problems
227 :return: `SolverOptions`
228 """
229 return self.__pde_p.getSolverOptions()
230
231 def setSolverOptionsPressure(self, options=None):
232 """
233 Sets the solver options used to solve the pressure problems
234 If ``options`` is not present, the options are reset to default
235
236 :param options: `SolverOptions`
237 :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
238 """
239 return self.__pde_p.setSolverOptions(options)
240
241 def solve(self, u0, p0):
242 """
243 solves the problem.
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. `escript.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. `escript.Data`).
249 :return: flux and pressure
250 :rtype: ``tuple`` of `escript.Data`.
251
252 """
253 p0=util.interpolate(p0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
254 if self.ref_point_id == None:
255 p_ref=0
256 else:
257 p_ref=p0.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
258 p0_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
259 g_2=self.__g - util.tensor_mult(self.__permeability, self.__permeability_invXg_ref * self.perm_scale)
260 self.__pde_p.setValue(X=g_2 * 1./self.perm_scale,
261 Y=self.__f * 1./self.perm_scale,
262 y= - util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux * 1./self.perm_scale ),
263 r=p0 - p0_hydrostatic)
264 pp=self.__pde_p.getSolution()
265 u = self._getFlux(pp, u0)
266 return u,pp + p0_hydrostatic
267
268 def getFlux(self,p, u0=None):
269 """
270 returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
271 on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
272 Notice that ``g`` is used, see `setValue`.
273
274 :param p: pressure.
275 :type p: scalar value on the domain (e.g. `escript.Data`).
276 :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
277 :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
278 :return: flux
279 :rtype: `escript.Data`
280 """
281 p=util.interpolate(p, self.__pde_p.getFunctionSpaceForCoefficient("q"))
282 if self.ref_point_id == None:
283 p_ref=0
284 else:
285 p_ref=p.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
286 p_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
287 return self._getFlux(p-p_hydrostatic, u0)
288
289 def _getFlux(self, pp, u0=None):
290 """
291 returns the flux for a given pressure ``pp`` where the flux is equal to
292 ``u0`` on locations where ``location_of_fixed_flux`` is positive (see
293 `setValue`). Notice that ``g`` is used, see `setValue`.
294
295 :param pp: pressure.
296 :type pp: scalar value on the domain (i.e. `escript.Data`).
297 :param u0: flux on the locations of the domain marked in ``location_of_fixed_flux``.
298 :type u0: vector values on the domain (i.e. `escript.Data`) or ``None``
299 :return: flux
300 :rtype: `escript.Data`
301 """
302 if self.solver == self.EVAL:
303 u = self.__g - util.tensor_mult(self.__permeability, self.perm_scale * (util.grad(pp) + self.__permeability_invXg_ref))
304 elif self.solver == self.POST or self.solver == self.SMOOTH:
305 self.__pde_v.setValue(Y= self.__permeability_invXg - (util.grad(pp) + self.__permeability_invXg_ref))
306
307 if u0 == None:
308 self.__pde_v.setValue(r=escore.Data())
309 else:
310 if not isinstance(u0, escore.Data) : u0 = escore.Vector(u0, escore.Solution(self.domain))
311 self.__pde_v.setValue(r=1./self.perm_scale * u0)
312 u= self.__pde_v.getSolution() * self.perm_scale
313 return u
314
315 class StokesProblemCartesian(pdt.HomogeneousSaddlePointProblem):
316 """
317 solves
318
319 -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
320 u_{i,i}=0
321
322 u=0 where fixed_u_mask>0
323 eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
324
325 if surface_stress is not given 0 is assumed.
326
327 typical usage:
328
329 sp=StokesProblemCartesian(domain)
330 sp.setTolerance()
331 sp.initialize(...)
332 v,p=sp.solve(v0,p0)
333 sp.setStokesEquation(...) # new values for some parameters
334 v1,p1=sp.solve(v,p)
335 """
336 def __init__(self,domain,**kwargs):
337 """
338 initialize the Stokes Problem
339
340 The approximation spaces used for velocity (=Solution(domain)) and pressure (=ReducedSolution(domain)) must be
341 LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
342 with macro elements for the pressure.
343
344 :param domain: domain of the problem.
345 :type domain: `Domain`
346 """
347 pdt.HomogeneousSaddlePointProblem.__init__(self,**kwargs)
348 self.domain=domain
349 self.__pde_v=lpe.LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
350 self.__pde_v.setSymmetryOn()
351
352 self.__pde_prec=lpe.LinearPDE(domain)
353 self.__pde_prec.setReducedOrderOn()
354 self.__pde_prec.setSymmetryOn()
355
356 self.__pde_proj=lpe.LinearPDE(domain)
357 self.__pde_proj.setReducedOrderOn()
358 self.__pde_proj.setValue(D=1)
359 self.__pde_proj.setSymmetryOn()
360
361 def getSolverOptionsVelocity(self):
362 """
363 returns the solver options used solve the equation for velocity.
364
365 :rtype: `SolverOptions`
366 """
367 return self.__pde_v.getSolverOptions()
368 def setSolverOptionsVelocity(self, options=None):
369 """
370 set the solver options for solving the equation for velocity.
371
372 :param options: new solver options
373 :type options: `SolverOptions`
374 """
375 self.__pde_v.setSolverOptions(options)
376 def getSolverOptionsPressure(self):
377 """
378 returns the solver options used solve the equation for pressure.
379 :rtype: `SolverOptions`
380 """
381 return self.__pde_prec.getSolverOptions()
382 def setSolverOptionsPressure(self, options=None):
383 """
384 set the solver options for solving the equation for pressure.
385 :param options: new solver options
386 :type options: `SolverOptions`
387 """
388 self.__pde_prec.setSolverOptions(options)
389
390 def setSolverOptionsDiv(self, options=None):
391 """
392 set the solver options for solving the equation to project the divergence of
393 the velocity onto the function space of presure.
394
395 :param options: new solver options
396 :type options: `SolverOptions`
397 """
398 self.__pde_proj.setSolverOptions(options)
399 def getSolverOptionsDiv(self):
400 """
401 returns the solver options for solving the equation to project the divergence of
402 the velocity onto the function space of presure.
403
404 :rtype: `SolverOptions`
405 """
406 return self.__pde_proj.getSolverOptions()
407
408 def updateStokesEquation(self, v, p):
409 """
410 updates the Stokes equation to consider dependencies from ``v`` and ``p``
411 :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values to model parameters.
412 """
413 pass
414 def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
415 """
416 assigns new values to the model parameters.
417
418 :param f: external force
419 :type f: `Vector` object in `FunctionSpace` `Function` or similar
420 :param fixed_u_mask: mask of locations with fixed velocity.
421 :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
422 :param eta: viscosity
423 :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
424 :param surface_stress: normal surface stress
425 :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
426 :param stress: initial stress
427 :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
428 """
429 if eta !=None:
430 k=util.kronecker(self.domain.getDim())
431 kk=util.outer(k,k)
432 self.eta=util.interpolate(eta, escore.Function(self.domain))
433 self.__pde_prec.setValue(D=1/self.eta)
434 self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
435 if restoration_factor!=None:
436 n=self.domain.getNormal()
437 self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
438 if fixed_u_mask!=None:
439 self.__pde_v.setValue(q=fixed_u_mask)
440 if f!=None: self.__f=f
441 if surface_stress!=None: self.__surface_stress=surface_stress
442 if stress!=None: self.__stress=stress
443
444 def initialize(self,f=escore.Data(),fixed_u_mask=escore.Data(),eta=1,surface_stress=escore.Data(),stress=escore.Data(), restoration_factor=0):
445 """
446 assigns values to the model parameters
447
448 :param f: external force
449 :type f: `Vector` object in `FunctionSpace` `Function` or similar
450 :param fixed_u_mask: mask of locations with fixed velocity.
451 :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
452 :param eta: viscosity
453 :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
454 :param surface_stress: normal surface stress
455 :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
456 :param stress: initial stress
457 :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
458 """
459 self.setStokesEquation(f,fixed_u_mask, eta, surface_stress, stress, restoration_factor)
460
461 def Bv(self,v,tol):
462 """
463 returns inner product of element p and div(v)
464
465 :param v: a residual
466 :return: inner product of element p and div(v)
467 :rtype: ``float``
468 """
469 self.__pde_proj.setValue(Y=-util.div(v))
470 self.getSolverOptionsDiv().setTolerance(tol)
471 self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
472 out=self.__pde_proj.getSolution()
473 return out
474
475 def inner_pBv(self,p,Bv):
476 """
477 returns inner product of element p and Bv=-div(v)
478
479 :param p: a pressure increment
480 :param Bv: a residual
481 :return: inner product of element p and Bv=-div(v)
482 :rtype: ``float``
483 """
484 return util.integrate(util.interpolate(p,escore.Function(self.domain))*util.interpolate(Bv, escore.Function(self.domain)))
485
486 def inner_p(self,p0,p1):
487 """
488 Returns inner product of p0 and p1
489
490 :param p0: a pressure
491 :param p1: a pressure
492 :return: inner product of p0 and p1
493 :rtype: ``float``
494 """
495 s0=util.interpolate(p0, escore.Function(self.domain))
496 s1=util.interpolate(p1, escore.Function(self.domain))
497 return util.integrate(s0*s1)
498
499 def norm_v(self,v):
500 """
501 returns the norm of v
502
503 :param v: a velovity
504 :return: norm of v
505 :rtype: non-negative ``float``
506 """
507 return util.sqrt(util.integrate(util.length(util.grad(v))**2))
508
509
510 def getDV(self, p, v, tol):
511 """
512 return the value for v for a given p
513
514 :param p: a pressure
515 :param v: a initial guess for the value v to return.
516 :return: dv given as *Adv=(f-Av-B^*p)*
517 """
518 self.updateStokesEquation(v,p)
519 self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
520 self.getSolverOptionsVelocity().setTolerance(tol)
521 self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
522 if self.__stress.isEmpty():
523 self.__pde_v.setValue(X=p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
524 else:
525 self.__pde_v.setValue(X=self.__stress+p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
526 out=self.__pde_v.getSolution()
527 return out
528
529 def norm_Bv(self,Bv):
530 """
531 Returns Bv (overwrite).
532
533 :rtype: equal to the type of p
534 :note: boundary conditions on p should be zero!
535 """
536 return util.sqrt(util.integrate(util.interpolate(Bv, escore.Function(self.domain))**2))
537
538 def solve_AinvBt(self,p, tol):
539 """
540 Solves *Av=B^*p* with accuracy `tol`
541
542 :param p: a pressure increment
543 :return: the solution of *Av=B^*p*
544 :note: boundary conditions on v should be zero!
545 """
546 self.__pde_v.setValue(Y=escore.Data(), y=escore.Data(), X=-p*util.kronecker(self.domain))
547 out=self.__pde_v.getSolution()
548 return out
549
550 def solve_prec(self,Bv, tol):
551 """
552 applies preconditioner for for *BA^{-1}B^** to *Bv*
553 with accuracy ``self.getSubProblemTolerance()``
554
555 :param Bv: velocity increment
556 :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
557 :note: boundary conditions on p are zero.
558 """
559 self.__pde_prec.setValue(Y=Bv)
560 self.getSolverOptionsPressure().setTolerance(tol)
561 self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
562 out=self.__pde_prec.getSolution()
563 return out

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