6.4 Two-level methods
6.5.1 Forchheimer’s equation in 1D
Forchheimer’s equation is an extension of the Darcy equation for high flow rates, where the linear relation between the flow velocity and the gradient flow does not hold anymore.
In a one dimensional domainΩ:=(0, 1), the Forchheimer model is q(−λ(x)u(x)0))0=f(x) inΩ,
u(0)=uL and u(1)=uR, (6.5.1)
whereuL,uR∈R,λ(x) is a positive and bounded permeability field andq(y) :=sign(y)−1+
p1+4γ|y|
2γ ,
withγ>0. To discretize (6.5.1), we use the finite volume scheme described in detail in [59]. In our numerical experiments, we setλ(x)=2+cos(5πx),f(x)=50 sin(5πx)ex,γ=1, u(0)=1 andu(1)=e. The solution fieldu(x) and the force fieldf(x) are shown in Figure 6.2. We then study the convergence behaviour of the different methods. Figure 6.3 shows how the relative error decays for the different methods and for a decomposition into 20 subdomains (left panel) and 50 subdomains (right panel). The initial guess is equal to zero for all the methods. From Figure 6.3, it seems that the convergence of the RASPEN and SRASPEN methods is not affected by the number of subdomains. However, Figure 6.3 does not tell the whole story, as one should focus not only on the number of iterations, but also on the cost of each iteration. To compare the cost of an iteration of the RASPEN and SRASPEN methods, we have to distinguish two cases, that is if one solves the Jacobian system directly or with some Krylov methods, e.g GMRES. First suppose that we want to
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Newton Nonlinear RAS RASPEN SRASPEN
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Newton Nonlinear RAS RASPEN SRASPEN
Figure 6.3: Convergence behaviour for the Newton method, the nonlinear RAS method, the RASPEN method and the SRASPEN method applied to the Forchheimer’s equation.
On the left, the simulation refers to a decomposition into 20 subdomains while on the right we consider 50 subdomains. The mesh size ish=10−3and the overlap is 8h.
solve the Jacobian system with a direct method and thus we need to assemble and store the Jacobians. From the expressions in equation (6.2.18) we remark that the assembly of the Jacobian of the RASPEN method requiresN×Nv subdomain solves, where N is the number of subdomains andNv is the number of unknowns in volume. On the other hand, the assembly of the Jacobian of SRASPEN method requiresN×NSsolves, whereNS
is the number of unknowns on the substructures andNS¿Nv. Thus, while the assembly ofJF is prohibitive, it can still be affordable to assembleJFS. Further, the direct solution of the Jacobian system is feasible and cheap asJFShas sizeNS×NS. Suppose now that we solve the Jacobian systems with GMRES and we indicate withI(k) andIS(k) the number of GMRES iterations to solve the volume and substructured Jacobian systems at thek-th outer Newton’s iteration. Each GMRES iteration requiresNsubdomain solves which can be performed in parallel. In our numerical experiment we have observed that generally IS(k)≤I(k), withI(k)−IS(k)≈0, 1, 2, that is GMRES requires the same number of itera-tions or slightly less to solve the substructured Jacobian system compared to the volume one.
To compare the two methods fairly, we follow [59] and introduce the quantityL(n) which counts the number of subdomain solves performed by the two methods up to iterationn, taking into account the advantages of a parallel implementation. We setL(n)=Pn
k=1Lki n+ I(k), whereLki n is the maximum over the subdomains of the number of Newton’s iter-ations required to solve the local subdomain problems at iterationk. The number of linear solves performed by GMRES should beI(k)·N, but as theN linear solves can be performed in parallel, the total cost of GMRES corresponds approximately toI(k) linear solves. Figure 6.4 shows the error decay with respect toL(n). We note that the two meth-ods require approximately the same computational cost and SRASPEN is slightly faster.
For the decomposition into 50 subdomains, the RASPEN method requires on average 91.5 GMRES iterations per Newton iteration, while the SRASPEN method requires an average
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RASPEN SRASPEN
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RASPEN SRASPEN
Figure 6.4: Relative error decay for the RASPEN method and the SRASPEN method applied to the Forchheimer equation with respect to the number of linear solves. On the left, the simulation refers to a decomposition into 20 subdomains while on the right we consider 50 subdomains. The mesh size ish=10−3.
of 90.87 iterations. The size of the substructured spaceVSisNS=98. For the decomposi-tion into 20 subdomains, the RASPEN method requires an average of 40 GMRES iteradecomposi-tions per Newton’s iteration, while the SRASPEN method needs 38 iterations. The size ofVSis NS=38, which means that GMRES reaches the given tolerance of 10−12after exactlyNS
steps which is the size of the substructured Jacobian. Under these circumstances, it can be convenient to actually assembleJFS, as it requiresNS·N subdomain solves which is the total cost of GMRES. Furthermore, theNS·N subdomain solves are embarrassingly par-allel, while theNS·N solves of GMRES can be parallelized in the spatial direction, but not in the iterative one. As a future work, we believe it will be interesting to study the conver-gence of a Quasi-Newton method based on the SRASPEN method, where one assembles the Jacobian substructured matrix after every few outer Newton iterations, reducing the overall computational cost.
As a final remark, we specify that Figure 6.4 has been obtained setting a zero initial guess for the nonlinear subdomain problems. However, at the iterationkof the RASPEN method one can use the subdomain restriction of the updated volume solution, that isRjuk−1, which has been obtained by solving the volume Jacobian system at iterationk−1 and is thus generally a better initial guess for the next iteration. On the other hand in the SRASPEN method, one could use the subdomain solutions computed at iterationk−1, i.e.
uik−1, as initial guesses for the nonlinear subdomain problems, as the substructured Jaco-bian system corrects only the substructured values. Numerical experiments showed that with this particular choice of initial guesses for the nonlinear subdomain problems, the SRASPEN method requires generally more Newton iterations to solve the local problems.
In this setting, there is not a method which is constantly faster than the other as it depends on a delicate trade-off between the better GMRES performance and the need to perform more Newton iterations for the nonlinear local problems in the SRASPEN method.
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Newton Iterative RAS RASPEN SRASPEN
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Newton Iterative RAS RASPEN SRASPEN
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Figure 6.5: Relative error decay versus the number of iterations (top row) and error decay versus number of linear solves (bottom row). The left figures refer to a decomposition into 4 subdomains, while the right figures to a decomposition into 25 subdomains. The mesh size ish=0.012 and the overlap is 8h.