Volume 9, Issue 3

We consider the the interpolation problem between and , where Ω is a polygonal domain in and is the subspace of functions in H 1 (Ω) which vanish on the Dirichlet part ( ∂ Ω) D of the boundary of Ω. The main result is that the interpolation spaces and coincide. An application of this result to a nonconforming finite element problem is presented.

A new functional framework for consistently stabilized discrete approximations to convection-diffusion problems was recently proposed. The key ideas are the evaluation of the residual in an inner product of the type H –1/2 and the realization of this inner product via explicitely computable multilevel decompositions of function spaces. Here we improve such approach, by taking into account the anisotropic nature of the convection-diffusion operator. We derive uniform (in the diffusion parameter) anisotropic estimates for both the exact and the discrete solutions and we study the convergence of the approximation. To this end we develop a functional framework involving anisotropic Sobolev spaces which depend on the velocity field.

Convergence rate estimate is given for an overlapping domain decomposition method for obstacle problems. It is shown that the computed solution will converge monotonically to the true solution if the intial value is above the obstacle and below the true solution. Moreover, the convergence rate is of the same order as the linear elliptic problems. Numerical experiments are shown both for the additive and the multiplicative Schwarz methods. If the overlapping size is increased by a factor of 2, the iteration number is reduced by a factor of 2.