A well-known theoretical foundation of a fictitious domain, also called embedding, method is a classical estimate that guaranties convergence of the solution of the fictitious domain problem to the solution of the original problem. One typical proof technique of the estimate utilizes a reduction of the problem to the interface of the fictitious domain extension, using a transmission condition. An analogous approach appears in studying domain decomposition methods without overlap, reducing the investigation to the surface that separates the subdomains.

On a continuous level, this analysis is usually performed in an $H^{1/2}$ norm for second order elliptic equations. This norm appears naturally for Poincare-Steklov operators, which are convenient to employ to formulate the transmission condition. Using recent advances in a regularity theory of Poincare-Steklov operators for Lipschitz domains, we provide, in the present paper, a similar analysis in an $H^{1/2+\alpha}$ norm with $\alpha > 0,$ for a simple model problem. As such, we take a homogeneous Neumann boundary value problem for the Laplacian, transformed into a diffusion equations with the diffusion coefficient one in the original domain and a small positive constant value in the fictitious domain. This result leads to a convergence theory of the fictitious domain method for a second order elliptic PDE in an $H^{1+\alpha}$ norm, while the classical result is in an $H^1$ norm. Here, $\alpha < 1/2$ for the case of Lipschitz domains we consider.

The talk is partially based on the paper:
A.V. Knyazev and Olof Widlund, " Lavrentiev Regularization + Ritz Approximation = Uniform Finite Element Error Estimates for Differential Equations with Rough Coefficients." Mathematics of Computation, posted on July 13, 2001, S0025-5718-01-01378-3 (to appear in print).