Wavelet and finite element solutions for the Neumann problem using fictitious domains

This paper presents a new fictitious domain formulation for the solution of a strongly elliptic boundary value problem with Neumann boundary conditions for a bounded domain in a finite-dimensional Euclidean space with a smooth (possibly only Lipschitz) boundary. This extends the domain to a larger rectangular domain with periodic boundary conditions for which fast solvers are available. The extended solution converges on the original domain in the appropriate function spaces as the penalty parameter approaches zero. Both wavelet-Galerkin and finite elements numerical approximation schemes are developed using this methodology. The convergence rates of both schemes are comparable, and the use of finite elements requires a parameterization of the boundary, while the wavelet-Galerkin method admits an implicit description of the boundary in terms of a wavelet representation of the boundary measure defined as the distributional gradient of the characteristic function of the interior. The accuracy of both methods is investigated and compared, both theoretically and for numerical test cases. The conclusion is that the methods are comparable, and that the wavelet method allows the use of more general boundaries which are not explicitly parametrized, which would be of greater advantage in higher dimensions (the numerical tests are carried out in two dimensions).