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new DarcyFlux solver
1 \section{Darcy Flux}
2 \label{DARCY FLUX}
3 We want to calculate the velocity $u$ and pressure $p$ on a domain $\Omega$ solving
4 the Darcy flux problem \index{Darcy flux}\index{Darcy flow}
5 \begin{equation}\label{DARCY PROBLEM}
6 \begin{array}{rcl}
7 u\hackscore{i} + \kappa\hackscore{ij} p\hackscore{,j} & = & g\hackscore{i} \\
8 u\hackscore{k,k} & = & f
9 \end{array}
10 \end{equation}
11 with the boundary conditions
12 \begin{equation}\label{DARCY BOUNDARY}
13 \begin{array}{rcl}
14 u\hackscore{i} \; n\hackscore{i} = u^{N}\hackscore{i} \; n\hackscore{i} & \mbox{ on } & \Gamma\hackscore{N} \\
15 p = p^{D} & \mbox{ on } & \Gamma\hackscore{D} \\
16 \end{array}
17 \end{equation}
18 where $\Gamma\hackscore{N}$ and $\Gamma\hackscore{D}$ are a partition of the boundary of $\Omega$ with $\Gamma\hackscore{D}$ non empty, $n\hackscore{i}$ is the outer normal field of the boundary of $\Omega$, $u^{N}\hackscore{i}$ and $p^{D}$ are given functions on $\Omega$, $g\hackscore{i}$ and $f$ are given source terms and $\kappa\hackscore{ij}$ is the given permeability. We assume that $\kappa\hackscore{ij}$ is symmetric (which is not really required) and positive definite, i.e there are positive constants $\alpha\hackscore{0}$ and $\alpha\hackscore{1}$ which are independent from the location in $\Omega$ such that
19 \begin{equation}
20 \alpha\hackscore{0} \; x\hackscore{i} x\hackscore{i} \le \kappa\hackscore{ij} x\hackscore{i} x\hackscore{j} \le \alpha\hackscore{1} \; x\hackscore{i} x\hackscore{i}
21 \end{equation}
22 for all $x\hackscore{i}$.
24 \subsection{Solution Method \label{DARCY SOLVE}}
25 It is useful to write equation~\ref{DARCY PROBLEM} in operator form. For any pressure $p$
26 we set
27 \begin{equation}
28 (Gp)\hackscore{i} = p\hackscore{,j}
29 \end{equation}
30 and a velocity $v$ we set
31 \begin{equation}
32 Dv = v\hackscore{k,k}
33 \end{equation}
34 We $K=(\kappa\hackscore{ij})$ we can write the Darcy problem~\ref{DARCY PROBLEM} as
35 \begin{equation}
36 \begin{array}{rcl}
37 u + K \, Gp & = & g \\
38 Du & = & f
39 \end{array}
40 \end{equation}
41 We solve this equation by minimising the functional
42 \begin{equation}\label{DARCY COST}
43 J(u,p):=\|K^{-\frac{1}{2}}(u + K \, G p - g)\|^2\hackscore{0} + \|\lambda (Du-f) \|\hackscore{0}^2
44 \end{equation}
45 over all suitable $u$ and $p$. In this equation we set $\|p\|^2\hackscore{0}=(p,p)\hackscore{0}$ with
46 \begin{equation}
47 (p,q)\hackscore{0} = \int\hackscore{\Omega } p\cdot q \, dx
48 \end{equation}
49 The factor $\lambda>0$ is a weighting factor.
50 A simple calculation shows that
51 one has to solve
52 \begin{equation}
53 ( K^{-1} (v + K \, Gq) , u +K \, G p - g)\hackscore{0} + (\lambda Dv,\lambda (Du-f) )\hackscore{0} =0
54 \end{equation}
55 for all velocities $v$ and pressure $q$ which fulfill the homogeneous boundary conditions~\ref{DARCY BOUNDARY}.
56 This so-called variational equation can be translated back into operator notation
57 \begin{equation}
58 \begin{array}{rcl}
59 (K^{-1}+ D^*\lambda^2 D)u + Gp & = & D^*\lambda f + K^{-1} g \\
60 G^*u + G^*K \, G p & = & G^*g \\
61 \end{array}
62 \end{equation}
63 where $D^*$ and $Q^*$ denote the adjoint operators with respect to $(.,.)\hackscore{0}$.
64 In~\cite{LEASTSQUARESFEM1994} it has been shown that this problem is continuous and coercive and therefore has a unique solution. Also standard FEM methods can be used for discretization. It is also possible
65 to solve the problem in coupled form, however this approach leads in some cases to a very ill-conditioned stiffness matrix in particular in the case of a very small or large permeability ($\alpha\hackscore{1} \ll 1$ or $\alpha\hackscore{0} \gg 1$)
67 The approach we are taking is to eliminate the velocity $v$ from the problem. Assuming that $p$ is known we have
68 \begin{equation}\label{DARCY V FORM}
69 v= (K^{-1}+ D^*\lambda^2 D)^{-1} ( D^*\lambda f + K^{-1} g - Gp)
70 \end{equation}
71 (notice that $K^{-1}+\lambda D^*D$ is coercive) which is inserted into the second equation
72 \begin{equation}
73 G^* (K^{-1}+\lambda D^*D)^{-1} (\lambda D^*f + K^{-1} g - Gp) + G^* KG p = G^*g
74 \end{equation}
75 which is
76 \begin{equation}
77 G^* ( K - (K^{-1}+ D^*\lambda^2 D)^{-1} ) G p = G^* (g-(K^{-1}+D^*\lambda^2 D)^{-1} ( D^*\lambda f + K^{-1} g) )
78 \end{equation}
79 We use the PCG \index{linear solver!PCG}\index{PCG} method to solve this.
80 The residual $r$ is given as
81 \begin{equation}
82 \begin{array}{rcl}
83 r & =& G^* \left( g - K\, G p - v \right)
84 \end{array}
85 \end{equation}
86 for the current pressure approximation $p$ and current velocity $v$ defined by
87 equation~\ref{DARCY V FORM}.
88 So in a particular implementation we use $\hat{r}=g-K\, Gp-v$ to represent the residual.
89 The evaluation of the iteration operator for a given $p$ is then
90 returning $Qp+v$ where $v$ is the solution of
91 \begin{equation}\label{UPDATE W}
92 (K^{-1}+ D^*\lambda^2 D)v = Gp
93 \end{equation}
94 To derive a preconditioner we use the identity
95 \begin{equation}
96 \begin{array}{rcl}
98 G^* ( K - (K^{-1}+ D^*\lambda^2 D)^{-1} ) G & = & G^* (I - (I + K D^*\lambda^2 D)^{-1}) K G \\
99 & \approx & G^* (K D^*\lambda^2 D) K G \\
100 & = & G^* K ( D^* \lambda^2 D) K G \\
101 & \approx & G^* \frac{\lambda^2}{dx^2} K^2 G
102 \end{array}
103 \end{equation}
104 where $dx$ is the local mesh size and we use the approximation
105 \begin{equation}
106 D^* \lambda^2 D \approx \frac{\lambda^2}{dx^2}
107 \end{equation}
108 Therefore we use $G^* \frac{\lambda^2}{dx^2} K^2 G$ as a preconditioner.To evalute the preconditioner
109 we need to solve the equation
110 \begin{equation}\label{UPDATE P}
111 G^* \frac{\lambda^2}{dx^2} K^2 G p = G^* \hat{r}
112 \end{equation}
113 It remains to answer the question how to choose $\lambda$. We need to balance the first and second
114 term in $J(u,p)$ in equation~\ref{DARCY COST}. We inspect $J$ for
115 $(\hat{u}, \hat{p})$ which is a perturbed exact solution $(u,p)$.
116 Assuming $\hat{u}=u+u\hackscore{0}e^{ik^tx}$
117 and $\hat{p}=p+p\hackscore{0}e^{ik^tx}$ and constant $K$ we get
118 \begin{equation}
119 J(\hat{u},\hat{p}) = C \left[ ( \|K^{-1}\|\hackscore{2} |u\hackscore{0}|^2 + \|K\|\hackscore{2} \|k\|\hackscore{2}^2 |p\hackscore{0}|^2| )
120 + \lambda^2 \|k\|\hackscore{2}^2 |u\hackscore{0}|^2 \right]
121 \end{equation}
122 with some constant $C>0$. The first two terms and the third term correspond to the first term and
123 second term in the definition of $J(u,p)$ in equation~\ref{DARCY COST}. For small $\|k\|\hackscore{2}$
124 (i.e. for a smooth perturbation) $J(\hat{u},\hat{p})$ is dominated by $\|K^{-1}\|\hackscore{2} |u\hackscore{0}|^2$.
125 To scale the second term which is corresponds to the incompressibility condition for the velocity
126 we need to meet the condition $\|K^{-1}\|\hackscore{2} = \lambda^2 \|k\|\hackscore{2}^2$.
127 Taking the boundary conditions into consideration the smallest possible value for $\|k\|\hackscore{2} = \frac{\pi}{l}$ where $l$ is the longest edge of the domain. This leads to the
128 \begin{equation}\label{DARCY LAMBDA}
129 \lambda = \|K^{-1}\|\hackscore{2}^{\frac{1}{2}} \frac{l}{\pi}
130 \end{equation}
131 Notice that with this setting the preconditioner $G^* \frac{\lambda^2}{dx^2} K^2 G$ becomes
132 equivalent to $G^* K G$ if $K$ is a diagonal matrix and the mesh has a constant mesh size.
134 The residual norm used in the PCG is given as
135 \begin{equation}\label{DARCY R NORM}
136 \|r\|\hackscore{PCG}^2 = \int r (G^* \frac{\lambda^2}{dx^2} K^2 G)^{-1} r \; dx =\int \hat{r} G ( G^* \frac{\lambda^2}{dx^2} K^2 G)^{-1} G^* \hat{r} \; dx \approx
137 \int \hat{r} K^{-1} \hat{r} \; dx
138 \end{equation}
139 The iteration is terminated if
140 \begin{equation}\label{DARCY STOP}
141 \|r\|\hackscore{PCG} \le \mbox{ATOL}
142 \end{equation}
143 where we set
144 \begin{equation}\label{DARCY ATOL DEF}
145 \mbox{ATOL} = \mbox{atol} + \mbox{rtol} \cdot \left(\frac{1}{\|K^{-\frac{1}{2}}v\|\hackscore{0}} + \frac{1}{\|K^{\frac{1}{2}} G p\|\hackscore{0}} \right)^{-1}
146 \end{equation}
147 where rtol is a given relative tolerance and $\mbox{atol}$ is a given absolute tolerance (typically $=0$).
148 Notice that if $Gp$ and $v$ both are zero, the pair $(0,p)$ is a solution.
149 The problem is that ATOL is depending on the solution $p$ and $v$ calculated form~\ref{DARCY V FORM}.
150 In practice one use the initial guess for $p$
151 to get a first value for ATOL. If the stopping criterion is met in the PCG iteration, a new $v$ is calculated from the current pressure approximation and ATOL is recalculated. If \ref{DARCY STOP} is still fulfilled the calculation is terminated and $(v,p)$ is returned. Otherwise PCG is restarted with a new ATOL.
153 \subsection{Functions}
154 \begin{classdesc}{DarcyFlow}{domain}
155 opens the Darcy flux problem\index{Darcy flux} on the \Domain domain.
156 \end{classdesc}
158 \begin{methoddesc}[DarcyFlow]{setValue}{\optional{f=None, \optional{g=None, \optional{location_of_fixed_pressure=None, \optional{location_of_fixed_flux=None,
159 \\\optional{permeability=None}}}}}}
160 assigns values to the model parameters. Values can be assigned using various calls - in particular
161 in a time dependent problem only values that change over time needs to be reset. The permeability can be defined as scalar (isotropic), a vector (orthotropic) or a matrix (anisotropic).
162 \var{f} and \var{g} are the corresponding parameters in~\ref{DARCY PROBLEM}.
163 The locations and components where the flux is prescribed are set by positive values in
164 \var{location_of_fixed_flux}.
165 The locations where the pressure is prescribed are set by
166 by positive values of \var{location_of_fixed_pressure}.
167 The values of the pressure and flux are defined by the initial guess.
168 Notice that at any point on the boundary of the domain the pressure or the normal component of
169 the flux must be defined. There must be at least one point where the pressure is prescribed.
170 The method will try to cast the given values to appropriate
171 \Data class objects.
172 \end{methoddesc}
174 \begin{methoddesc}[DarcyFlow]{setTolerance}{\optional{rtol=1e-4}}
175 sets the relative tolerance \mbox{rtol} in \ref{DARCY ATOL DEF}.
176 \end{methoddesc}
178 \begin{methoddesc}[DarcyFlow]{setAbsoluteTolerance}{\optional{atol=0.}}
179 sets the absolute tolerance \mbox{atol} in \ref{DARCY ATOL DEF}.
180 \end{methoddesc}
182 \begin{methoddesc}[DarcyFlow]{getSolverOptionsFlux}{}
183 Returns the solver options used to solve the flux problems~(\ref{DARCY V FORM}) and~(\ref{UPDATE W}). Use the returned \SolverOptions object to control the solution algorithms.
184 \end{methoddesc}
186 \begin{methoddesc}[DarcyFlow]{getSolverOptionsPressure}{}
187 Returns the solver options used to solve the pressure problems~(\ref{UPDATE P}) as a preconditioner.
188 Use the returned \SolverOptions object to control the solution algorithms.
189 \end{methoddesc}
191 \begin{methoddesc}[DarcyFlow]{solve}{u0,p0, \optional{max_iter=100, \optional{verbose=False}}}
192 solves the problem. and returns approximations for the flux $v$ and the pressure $p$.
193 \var{u0} and \var{p0} define initial guess for flux and pressure. Values marked
194 by positive values \var{location_of_fixed_flux} and \var{location_of_fixed_pressure}, respectively, are kept unchanged. \var{max_iter} sets the maximum number of iterations steps allowed for solving the coupled problem.
195 \end{methoddesc}
198 \subsection{Example: Gravity Flow}
199 later

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