Coupled FETI/BETI solvers for nonlinear potential problems in (un ...

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4 Ulrich Langer and Clemens Pechstein dimensions the single layer potential operator V i is always elliptic, whereas in two dimensions, due to the logarithm in the fundamental solution, it is only elliptic if diam Ω i < 1. This property can always be achieved by a suitable coordinate scaling. After discretizing and eliminating t i, one obtains the symmetric and stable approximation Si, BEM h := D i, h + ` α i 2 Mi, ⊤ h + Ki, ⊤ −1` h ´V αi i, h 2 M i, h + K i, h´ , (8) where V i, h , K i, h , D i, h are the boundary element matrices corresponding to V i, K i, D i, respectively, and M i, h is a mass matrix. Note that the two approximations Si, FEM h and Si, BEM h are compatible and both spectrally equivalent to the Galerkin matrices of the exact Steklov-Poincaré operators S i. The application of Si, FEM h or Si, BEM h simply corresponds to the solution of local Dirichlet problems. For details we refer to [11, 12]. 2.2 Tearing and Interconnecting Introducing separate variables u i on the local subdomains, one can re-enforce the continuity of the solution u across interfaces Γ ij by the constraints X p Bi ui = 0 , (9) i=1 where the B i are incidence matrices. Problem (2) can be written as a constraint minimization problem, as well as a saddle point problem involving Lagrange multipliers. Using the notion of the pseudoinverse ( † ) and a special projection P addressing the kernels of the subproblems, it is possible to eliminate the primal unknowns u i. Finally, one obtains the discrete dual FETI/BETI formulation, to find the Lagrange multiplier λ such that where the FETI/BETI operator F is defined by F = B where B = ˆB i˜p , i=1 SFEM/BEM h := P T F λ = d, (10) FEM/BEM˜† ˆSh B ⊤ = X p ˜† Bi ˆSFEM/BEM i=1 i,h Bi ⊤ , (11) FEM/BEM˜p ˆSi, h . The application of the pseudo- i=1 inverses [S FEM/BEM i, h ] † can be realized by the simple solution of regularized local Neumann problems. Since F is symmetric positive definite on range(P), one can solve the dual problem (10) by a preconditioned conjugate gradient subspace iteration. The preconditioner M −1 S,α = (BD −1 α B ⊤ ) −1 BD α S FEM/BEM h D αB ⊤ (BD −1 α B ⊤ ) −1 , (12) first introduced and fully analyzed by Klawonn and Widlund [7], satisfies the quasioptimal condition number estimate κ(PM −1 S,αP T P T FP) ≤ C(1 + log(H/h)) 2 , (13) independent of the values–and therefore possible jumps–of the coefficients α i. It is well known that an appropriate norm of the √iteration error of the conjugate gradient method will decrease at least by a factor 2` κ−1 ´n √ κ+1 in n steps. The robustness with
Coupled FETI/BETI solvers – nonlinear, unbounded 5 respect to the jumps in the coefficients α i is due to the special diagonal scaling matrix D α, involving weighted mean values of α i on cross points, interface lines and interfaces between the subdomains. We further point out that one step of the PCG subspace iteration is performed by the solution of one local Dirichlet and one local Neumann problem on each of the subdomains, and the application of the projection and some scaling matrices, which are both global operations but of a rather small dimension. This is why these tearing and interconnecting methods are most suitable for parallelization. For a more detailed description see e. g. [7, 10, 12] 2.3 Varying coefficients In this subsection, we consider a varying matrix coefficient A i(x) instead of a constant coefficient α i on some of the subdomains discretized by the FEM. We assume A i(x) to be constant on the finite elements T ∈ T i, h . In order to determine the amount of variance, we introduce the spectral variance measure m SV (A i) := sup x∈Ω i α i (x) inf x∈Ωi α i (x) , (14) where α i(x) and α i (x) denote the maximal and minimal local eigenvalues of A i(x), respectively. The application of a preconditioner with Steklov-Poincaré operators corresponding to constant coefficients leads in the worst case to a condition number proportional to max i m SV (A i), which is not at all acceptable in magnetostatic applications. In [10] the present authors have proposed a new preconditioner M c−1 S,A based on a varying scalar coefficient bα i(x) which can be easily computed from A i(x), together with a suitable diagonal scaling matrix D bα . If the local anisotropy measure m anis(A i) := sup x∈Ωi α i (x) α i (x) (15) is moderate, our new preconditioner works fine, cf. Table 1 and [10]. 3 Nonlinear Problems We now consider the following nonlinear magnetostatic model problem: −div[ν i(|∇u|)∇u] = f in Ω i , u = 0 on Γ , ν i(|∇u|) ∂u ∂n i + ν j(|∇u|) ∂u ∂n j = 0 on Γ ij . Assuming that the functions t ↦→ ν i(t)t : [0, ∞) → [0, ∞) are strongly monotonically increasing and piecewise C 2 , (16) is uniquely solvable in the weak sense and the corresponding Newton iteration converges locally at a quadratic rate. For our computations, we generated these material curves from noisy measurements using the robust interproximation technique introduced by Pechstein and Jüttler in [15]. In the linearized problems a varying matrix coefficient ζ i(∇u (k) h (x)) appears. In our numerical experiments, it turns out that typically the anisotropy measure m anis(ζ i(∇u (k) h )) is small, whereas the spectral variance measure m SV(ζ i(∇u (k) h )) becomes rather large. As one can observe in Table 1, with our new preconditioner cM −1 S,A such linearized Newton-problems can be solved satisfactorily. In order to get a good initial guess for Newton’s iteration, it is convenient to set up a hierarchy of nested grids and use coarse grid solutions as initial guesses on finer levels. For details, see [10]. (16)
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