Overlapping DD preconditioners for degenerated problems

Priv.-Doz. Dr. Sven Beuchler

Nov. 7, 2006, 3:30 p.m. T 1010

In this talk, we investigate problems of the type $-\nabla \cdot D \nabla u = f$ in $\Omega = (0,1)^2$. The matrix $D$ is bounded and positive definite, but not uniformly positive definite. Such problems are called degenerated problems. We consider two cases for the diffusion matrix $D$

$\begin{eqnarray}I: D & = & \omega^2(x)I, \mathrm{where } I \mathrm{ is the unity matrix}, \\ II: D & = & \left[\begin{array}{cc}\omega_2^2(y) & 0 \\ 0 & \omega_1^2(x) \end{array} \right], \end{eqnarray}$

with weight functions of the type $\omega(\xi) = \xi^\alpha, \alpha \ge 0$. Examples of degenerated problems are the elliptic part of the Black-Scholes pde (Case I with $\omega^2(x) = x^2$) and the solver related to the interior bubbles for the p-version of the finite element method (Case II with $\omega_i(\xi) = \xi,i = 1,2$). The problems are discretized by piecewise linear triangular elements on the regular grids. The linear system is solved by a pcg-method with an overlapping Domain Decomposition preconditioner. The optimality of the solution method is proved. Numerical Experiments show the performance of the proposed method.

This is joint work with Prof. Sergey Nepomnyaschikh (Novosibirsk).