Transmission on Lipschitz boundaries in stronger norms Analysis of transmission problems on Lipschitz boundaries in stronger norms

Abstract For a model diffusion equation on a Lipschitz simply connected bounded domain with a small diffusion coefficient in a Lipschitz simply connected subdomain located strictly inside of the original domain, Knyazev [2003] studies asymptotic properties of the solution with respect to the small diffusion coefficient vanishing. This problem asymptotically turns into a diffusion problem with a cavity. One proof technique of this fact utilizes a reduction of the problem to the boundary of the inner subdomain, 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 and in theoretical foundation of a fictitious domain method, e.g., to prove a classical estimate that guaranties convergence of the solution of the fictitious domain problem to the solution with the cavity. Transmission on Lipschitz boundaries in stronger norms Andrew Knyazev, University of Colorado at Denver 3 Center for Computational Mathematics, University of Colorado at Denver On a continuous level, the analysis is usually performed in a trace norm, e.g., the H 1/2 norm for second order elliptic equations. This norm appears naturally for Poincaré-Steklov operators, which are convenient to employ to formulate the transmission condition. A regularity theory of Poincaré-Steklov operators for Lipschitz domains is closely related to regularity of transmission and diffraction problems and properties of layer potentials on nonsmooth interfaces, and has been extensively studied. Taking advantage of the recent progress in the regularity theory for Lipschitz domains, Knyazev [2003] provides an analysis of the transmission condition in the H 1/2+α norm with α > 0, for a simple model problem. This result leads to a convergence theory of the fictitious domain method for a second order elliptic PDE in an H 1+α norm, while the standard result is in an H 1 norm. For the case of Lipschitz domains that we consider, α < 1/2. This is an extension to Lipschitz domains of a classical result Kopčenov [1974] for smooth domains. Transmission on Lipschitz boundaries in stronger norms Andrew Knyazev, University of Colorado at Denver 4 Center for Computational Mathematics, University of Colorado at Denver Knyazev [2003] proof relies on the regularity of the solution of the diffusion equation uniformly in the value of the small diffusion coefficient in the subdomain, established in Knyazev and Widlund [2003]. The solution is regular away from the boundary and the interface of the jump in the diffusion coefficient for a sufficiently smooth right-hand side. Under our simplifying assumption that the interface of the jump is strictly inside, the question of regularity can be reduced to studying layer potentials on the interface without boundary conditions, since the interface is then is a closed Lipschitz curve without selfintersection and it does not have any common (junction) points with the boundary, where the homogeneous Dirichlet boundary condition is enforced. We show in Knyazev and Widlund [2003] that the solution belongs to the space H 1/2+α norm with 1/2 > α > 0, uniformly in the small diffusion coefficient vanishing in the inner subdomain. Transmission on Lipschitz boundaries in stronger norm