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Round 1 of copyright fixes
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
37 escore=escriptcpp
38 #from . import escript
39 from . import util
40 from .linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
41 from .pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
42
43 class DarcyFlow(object):
44 """
45 solves the problem
46
47 *u_i+k_{ij}*p_{,j} = g_i*
48 *u_{i,i} = f*
49
50 where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
51
52 :cvar EVAL: direct pressure gradient evaluation for flux
53 :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*
54 where *l* is the length scale, *K* is the inverse of the permeability tensor, and *w* is a positive weighting factor.
55 :cvar SMOOTH: global smoothing by solving the PDE *K_{ij} u_j= - p_{,j} + K_{ij} g_j*
56 """
57 EVAL="EVAL"
58 SIMPLE="EVAL"
59 POST="POST"
60 SMOOTH="SMOOTH"
61 def __init__(self, domain, useReduced=False, solver="POST", verbose=False, w=1.):
62 """
63 initializes the Darcy flux problem.
64
65 :param domain: domain of the problem
66 :type domain: `Domain`
67 :param useReduced: uses reduced oreder on flux and pressure
68 :type useReduced: ``bool``
69 :param solver: solver method
70 :type solver: in [`DarcyFlow.EVAL`, `DarcyFlow.POST`, `DarcyFlow.SMOOTH` ]
71 :param verbose: if ``True`` some information on the iteration progress are printed.
72 :type verbose: ``bool``
73 :param w: weighting factor for `DarcyFlow.POST` solver
74 :type w: ``float``
75
76 """
77 if not solver in [DarcyFlow.EVAL, DarcyFlow.POST, DarcyFlow.SMOOTH ] :
78 raise ValueError("unknown solver %d."%solver)
79
80 self.domain=domain
81 self.solver=solver
82 self.useReduced=useReduced
83 self.verbose=verbose
84 self.l=None
85 self.w=None
86
87 self.__pde_p=LinearSinglePDE(domain)
88 self.__pde_p.setSymmetryOn()
89 if self.useReduced: self.__pde_p.setReducedOrderOn()
90
91 if self.solver == self.EVAL:
92 self.__pde_v=None
93 if self.verbose: print("DarcyFlow: simple solver is used.")
94
95 elif self.solver == self.POST:
96 if util.inf(w)<0.:
97 raise ValueError("Weighting factor must be non-negative.")
98 if self.verbose: print("DarcyFlow: global postprocessing of flux is used.")
99 self.__pde_v=LinearPDESystem(domain)
100 self.__pde_v.setSymmetryOn()
101 if self.useReduced: self.__pde_v.setReducedOrderOn()
102 self.w=w
103 x=self.domain.getX()
104 self.l=min( [util.sup(x[i])-util.inf(x[i]) for i in range(self.domain.getDim()) ] )
105 #self.l=util.vol(self.domain)**(1./self.domain.getDim()) # length scale
106
107 elif self.solver == self.SMOOTH:
108 self.__pde_v=LinearPDESystem(domain)
109 self.__pde_v.setSymmetryOn()
110 if self.useReduced: self.__pde_v.setReducedOrderOn()
111 if self.verbose: print("DarcyFlow: flux smoothing is used.")
112 self.w=0
113
114 self.__f=escore.Data(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
115 self.__g=escore.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
116 self.__permeability_invXg=escore.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
117 self.__permeability_invXg_ref=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
118 self.ref_point_id=None
119 self.ref_point=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
120 self.location_of_fixed_pressure = escore.Data(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
121 self.location_of_fixed_flux = escore.Vector(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
122 self.perm_scale=1.
123
124
125 def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
126 """
127 assigns values to model parameters
128
129 :param f: volumetic sources/sinks
130 :type f: scalar value on the domain (e.g. `escript.Data`)
131 :param g: flux sources/sinks
132 :type g: vector values on the domain (e.g. `escript.Data`)
133 :param location_of_fixed_pressure: mask for locations where pressure is fixed
134 :type location_of_fixed_pressure: scalar value on the domain (e.g. `escript.Data`)
135 :param location_of_fixed_flux: mask for locations where flux is fixed.
136 :type location_of_fixed_flux: vector values on the domain (e.g. `escript.Data`)
137 :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
138 :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
139
140 :note: the values of parameters which are not set by calling ``setValue`` are not altered.
141 :note: at any point on the boundary of the domain the pressure
142 (``location_of_fixed_pressure`` >0) or the normal component of the
143 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
144 is along the *x_i* axis.
145
146 """
147 if location_of_fixed_pressure!=None:
148 self.location_of_fixed_pressure=util.wherePositive(util.interpolate(location_of_fixed_pressure, self.__pde_p.getFunctionSpaceForCoefficient("q")))
149 self.ref_point_id=self.location_of_fixed_pressure.maxGlobalDataPoint()
150 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.")
151 self.ref_point=self.__pde_p.getFunctionSpaceForCoefficient("q").getX().getTupleForGlobalDataPoint(*self.ref_point_id)
152 if self.verbose: print(("DarcyFlow: reference point at %s."%(self.ref_point,)))
153 self.__pde_p.setValue(q=self.location_of_fixed_pressure)
154 if location_of_fixed_flux!=None:
155 self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
156 if not self.__pde_v == None:
157 self.__pde_v.setValue(q=self.location_of_fixed_flux)
158
159 if permeability!=None:
160
161 perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
162 self.perm_scale=util.Lsup(util.length(perm))
163 if self.verbose: print(("DarcyFlow: permeability scaling factor = %e."%self.perm_scale))
164 perm=perm*(1./self.perm_scale)
165
166 if perm.getRank()==0:
167
168 perm_inv=(1./perm)
169 perm_inv=perm_inv*util.kronecker(self.domain.getDim())
170 perm=perm*util.kronecker(self.domain.getDim())
171
172
173 elif perm.getRank()==2:
174 perm_inv=util.inverse(perm)
175 else:
176 raise ValueError("illegal rank of permeability.")
177
178 self.__permeability=perm
179 self.__permeability_inv=perm_inv
180
181 #====================
182 self.__pde_p.setValue(A=self.__permeability)
183 if self.solver == self.EVAL:
184 pass # no extra work required
185 elif self.solver == self.POST:
186 k=util.kronecker(self.domain.getDim())
187 self.omega = self.w*util.length(perm_inv)*self.l*self.domain.getSize()
188 #self.__pde_v.setValue(D=self.__permeability_inv, A=self.omega*util.outer(k,k))
189 self.__pde_v.setValue(D=self.__permeability_inv, A_reduced=self.omega*util.outer(k,k))
190 elif self.solver == self.SMOOTH:
191 self.__pde_v.setValue(D=self.__permeability_inv)
192
193 if g != None:
194 g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
195 if g.isEmpty():
196 g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
197 else:
198 if not g.getShape()==(self.domain.getDim(),): raise ValueError("illegal shape of g")
199 self.__g=g
200 self.__permeability_invXg=util.tensor_mult(self.__permeability_inv,self.__g * (1./self.perm_scale ))
201 self.__permeability_invXg_ref=util.integrate(self.__permeability_invXg)/util.vol(self.domain)
202 if f !=None:
203 f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
204 if f.isEmpty():
205 f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
206 else:
207 if f.getRank()>0: raise ValueError("illegal rank of f.")
208 self.__f=f
209
210 def getSolverOptionsFlux(self):
211 """
212 Returns the solver options used to solve the flux problems
213 :return: `SolverOptions`
214 """
215 if self.__pde_v == None:
216 return None
217 else:
218 return self.__pde_v.getSolverOptions()
219
220 def setSolverOptionsFlux(self, options=None):
221 """
222 Sets the solver options used to solve the flux problems
223 If ``options`` is not present, the options are reset to default
224 :param options: `SolverOptions`
225 """
226 if not self.__pde_v == None:
227 self.__pde_v.setSolverOptions(options)
228
229 def getSolverOptionsPressure(self):
230 """
231 Returns the solver options used to solve the pressure problems
232 :return: `SolverOptions`
233 """
234 return self.__pde_p.getSolverOptions()
235
236 def setSolverOptionsPressure(self, options=None):
237 """
238 Sets the solver options used to solve the pressure problems
239 If ``options`` is not present, the options are reset to default
240
241 :param options: `SolverOptions`
242 :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
243 """
244 return self.__pde_p.setSolverOptions(options)
245
246 def solve(self, u0, p0):
247 """
248 solves the problem.
249
250 :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.
251 :type u0: vector value on the domain (e.g. `escript.Data`).
252 :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.
253 :type p0: scalar value on the domain (e.g. `escript.Data`).
254 :return: flux and pressure
255 :rtype: ``tuple`` of `escript.Data`.
256
257 """
258 p0=util.interpolate(p0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
259 if self.ref_point_id == None:
260 p_ref=0
261 else:
262 p_ref=p0.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
263 p0_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
264 g_2=self.__g - util.tensor_mult(self.__permeability, self.__permeability_invXg_ref * self.perm_scale)
265 self.__pde_p.setValue(X=g_2 * 1./self.perm_scale,
266 Y=self.__f * 1./self.perm_scale,
267 y= - util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux * 1./self.perm_scale ),
268 r=p0 - p0_hydrostatic)
269 pp=self.__pde_p.getSolution()
270 u = self._getFlux(pp, u0)
271 return u,pp + p0_hydrostatic
272
273 def getFlux(self,p, u0=None):
274 """
275 returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
276 on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
277 Notice that ``g`` is used, see `setValue`.
278
279 :param p: pressure.
280 :type p: scalar value on the domain (e.g. `escript.Data`).
281 :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
282 :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
283 :return: flux
284 :rtype: `escript.Data`
285 """
286 p=util.interpolate(p, self.__pde_p.getFunctionSpaceForCoefficient("q"))
287 if self.ref_point_id == None:
288 p_ref=0
289 else:
290 p_ref=p.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
291 p_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
292 return self._getFlux(p-p_hydrostatic, u0)
293
294 def _getFlux(self, pp, u0=None):
295 """
296 returns the flux for a given pressure ``pp`` where the flux is equal to
297 ``u0`` on locations where ``location_of_fixed_flux`` is positive (see
298 `setValue`). Notice that ``g`` is used, see `setValue`.
299
300 :param pp: pressure.
301 :type pp: scalar value on the domain (i.e. `escript.Data`).
302 :param u0: flux on the locations of the domain marked in ``location_of_fixed_flux``.
303 :type u0: vector values on the domain (i.e. `escript.Data`) or ``None``
304 :return: flux
305 :rtype: `escript.Data`
306 """
307 if self.solver == self.EVAL:
308 u = self.__g - util.tensor_mult(self.__permeability, self.perm_scale * (util.grad(pp) + self.__permeability_invXg_ref))
309 elif self.solver == self.POST or self.solver == self.SMOOTH:
310 self.__pde_v.setValue(Y= self.__permeability_invXg - (util.grad(pp) + self.__permeability_invXg_ref))
311 print
312 if u0 == None:
313 self.__pde_v.setValue(r=escore.Data())
314 else:
315 if not isinstance(u0, escore.Data) : u0 = escore.Vector(u0, escore.Solution(self.domain))
316 self.__pde_v.setValue(r=1./self.perm_scale * u0)
317 u= self.__pde_v.getSolution() * self.perm_scale
318 return u
319
320 class StokesProblemCartesian(HomogeneousSaddlePointProblem):
321 """
322 solves
323
324 -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
325 u_{i,i}=0
326
327 u=0 where fixed_u_mask>0
328 eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
329
330 if surface_stress is not given 0 is assumed.
331
332 typical usage:
333
334 sp=StokesProblemCartesian(domain)
335 sp.setTolerance()
336 sp.initialize(...)
337 v,p=sp.solve(v0,p0)
338 sp.setStokesEquation(...) # new values for some parameters
339 v1,p1=sp.solve(v,p)
340 """
341 def __init__(self,domain,**kwargs):
342 """
343 initialize the Stokes Problem
344
345 The approximation spaces used for velocity (=Solution(domain)) and pressure (=ReducedSolution(domain)) must be
346 LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
347 with macro elements for the pressure.
348
349 :param domain: domain of the problem.
350 :type domain: `Domain`
351 """
352 HomogeneousSaddlePointProblem.__init__(self,**kwargs)
353 self.domain=domain
354 self.__pde_v=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
355 self.__pde_v.setSymmetryOn()
356
357 self.__pde_prec=LinearPDE(domain)
358 self.__pde_prec.setReducedOrderOn()
359 self.__pde_prec.setSymmetryOn()
360
361 self.__pde_proj=LinearPDE(domain)
362 self.__pde_proj.setReducedOrderOn()
363 self.__pde_proj.setValue(D=1)
364 self.__pde_proj.setSymmetryOn()
365
366 def getSolverOptionsVelocity(self):
367 """
368 returns the solver options used solve the equation for velocity.
369
370 :rtype: `SolverOptions`
371 """
372 return self.__pde_v.getSolverOptions()
373 def setSolverOptionsVelocity(self, options=None):
374 """
375 set the solver options for solving the equation for velocity.
376
377 :param options: new solver options
378 :type options: `SolverOptions`
379 """
380 self.__pde_v.setSolverOptions(options)
381 def getSolverOptionsPressure(self):
382 """
383 returns the solver options used solve the equation for pressure.
384 :rtype: `SolverOptions`
385 """
386 return self.__pde_prec.getSolverOptions()
387 def setSolverOptionsPressure(self, options=None):
388 """
389 set the solver options for solving the equation for pressure.
390 :param options: new solver options
391 :type options: `SolverOptions`
392 """
393 self.__pde_prec.setSolverOptions(options)
394
395 def setSolverOptionsDiv(self, options=None):
396 """
397 set the solver options for solving the equation to project the divergence of
398 the velocity onto the function space of presure.
399
400 :param options: new solver options
401 :type options: `SolverOptions`
402 """
403 self.__pde_proj.setSolverOptions(options)
404 def getSolverOptionsDiv(self):
405 """
406 returns the solver options for solving the equation to project the divergence of
407 the velocity onto the function space of presure.
408
409 :rtype: `SolverOptions`
410 """
411 return self.__pde_proj.getSolverOptions()
412
413 def updateStokesEquation(self, v, p):
414 """
415 updates the Stokes equation to consider dependencies from ``v`` and ``p``
416 :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values to model parameters.
417 """
418 pass
419 def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
420 """
421 assigns new values to the model parameters.
422
423 :param f: external force
424 :type f: `Vector` object in `FunctionSpace` `Function` or similar
425 :param fixed_u_mask: mask of locations with fixed velocity.
426 :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
427 :param eta: viscosity
428 :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
429 :param surface_stress: normal surface stress
430 :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
431 :param stress: initial stress
432 :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
433 """
434 if eta !=None:
435 k=util.kronecker(self.domain.getDim())
436 kk=util.outer(k,k)
437 self.eta=util.interpolate(eta, escore.Function(self.domain))
438 self.__pde_prec.setValue(D=1/self.eta)
439 self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
440 if restoration_factor!=None:
441 n=self.domain.getNormal()
442 self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
443 if fixed_u_mask!=None:
444 self.__pde_v.setValue(q=fixed_u_mask)
445 if f!=None: self.__f=f
446 if surface_stress!=None: self.__surface_stress=surface_stress
447 if stress!=None: self.__stress=stress
448
449 def initialize(self,f=escore.Data(),fixed_u_mask=escore.Data(),eta=1,surface_stress=escore.Data(),stress=escore.Data(), restoration_factor=0):
450 """
451 assigns values to the model parameters
452
453 :param f: external force
454 :type f: `Vector` object in `FunctionSpace` `Function` or similar
455 :param fixed_u_mask: mask of locations with fixed velocity.
456 :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
457 :param eta: viscosity
458 :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
459 :param surface_stress: normal surface stress
460 :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
461 :param stress: initial stress
462 :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
463 """
464 self.setStokesEquation(f,fixed_u_mask, eta, surface_stress, stress, restoration_factor)
465
466 def Bv(self,v,tol):
467 """
468 returns inner product of element p and div(v)
469
470 :param v: a residual
471 :return: inner product of element p and div(v)
472 :rtype: ``float``
473 """
474 self.__pde_proj.setValue(Y=-util.div(v))
475 self.getSolverOptionsDiv().setTolerance(tol)
476 self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
477 out=self.__pde_proj.getSolution()
478 return out
479
480 def inner_pBv(self,p,Bv):
481 """
482 returns inner product of element p and Bv=-div(v)
483
484 :param p: a pressure increment
485 :param Bv: a residual
486 :return: inner product of element p and Bv=-div(v)
487 :rtype: ``float``
488 """
489 return util.integrate(util.interpolate(p,escore.Function(self.domain))*util.interpolate(Bv, escore.Function(self.domain)))
490
491 def inner_p(self,p0,p1):
492 """
493 Returns inner product of p0 and p1
494
495 :param p0: a pressure
496 :param p1: a pressure
497 :return: inner product of p0 and p1
498 :rtype: ``float``
499 """
500 s0=util.interpolate(p0, escore.Function(self.domain))
501 s1=util.interpolate(p1, escore.Function(self.domain))
502 return util.integrate(s0*s1)
503
504 def norm_v(self,v):
505 """
506 returns the norm of v
507
508 :param v: a velovity
509 :return: norm of v
510 :rtype: non-negative ``float``
511 """
512 return util.sqrt(util.integrate(util.length(util.grad(v))**2))
513
514
515 def getDV(self, p, v, tol):
516 """
517 return the value for v for a given p
518
519 :param p: a pressure
520 :param v: a initial guess for the value v to return.
521 :return: dv given as *Adv=(f-Av-B^*p)*
522 """
523 self.updateStokesEquation(v,p)
524 self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
525 self.getSolverOptionsVelocity().setTolerance(tol)
526 self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
527 if self.__stress.isEmpty():
528 self.__pde_v.setValue(X=p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
529 else:
530 self.__pde_v.setValue(X=self.__stress+p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
531 out=self.__pde_v.getSolution()
532 return out
533
534 def norm_Bv(self,Bv):
535 """
536 Returns Bv (overwrite).
537
538 :rtype: equal to the type of p
539 :note: boundary conditions on p should be zero!
540 """
541 return util.sqrt(util.integrate(util.interpolate(Bv, escore.Function(self.domain))**2))
542
543 def solve_AinvBt(self,p, tol):
544 """
545 Solves *Av=B^*p* with accuracy `tol`
546
547 :param p: a pressure increment
548 :return: the solution of *Av=B^*p*
549 :note: boundary conditions on v should be zero!
550 """
551 self.__pde_v.setValue(Y=escore.Data(), y=escore.Data(), X=-p*util.kronecker(self.domain))
552 out=self.__pde_v.getSolution()
553 return out
554
555 def solve_prec(self,Bv, tol):
556 """
557 applies preconditioner for for *BA^{-1}B^** to *Bv*
558 with accuracy ``self.getSubProblemTolerance()``
559
560 :param Bv: velocity increment
561 :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
562 :note: boundary conditions on p are zero.
563 """
564 self.__pde_prec.setValue(Y=Bv)
565 self.getSolverOptionsPressure().setTolerance(tol)
566 self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
567 out=self.__pde_prec.getSolution()
568 return out

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