Wavelet Techniques for the Fictitious-Domain-Lagrange-Multiplier-Approach

Abstract

We consider second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of Lagrange multipliers. This is combined with a fictitious domain approach into which the physical domain is embedded. The resulting saddle point problem will be discretized in terms of wavelets, resulting in an operator equation in l2. Stability of the discretization and consequently the uniform boundedness of the condition number of the finite-dimensional operator independent of the discretization is guaranteed by an appropriate LBB condition. For the iterative solution of the saddle point system, an incomplete Uzawa algorithm is employed. It can be shown that the iterative scheme combined with a nested iteration strategy is asymptotically optimal in the sense that it provides the solution up to discretization error on discretization level J in an overall amount of iterations of order O(N J ), where N J is the number of unknowns on level J. Finally, numerical results are provided.