Inverse estimates for elliptic boundary integral operators and their application to the adaptive coupling of FEM and BEM

We prove inverse-type estimates for the four classical boundary integral operators associated with the Laplace operator. These estimates are used to show convergence of an h-adaptive algorithm for the coupling of a finite element method with a boundary element method which is driven by a weighted residual error estimator

Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems

We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear) interface problem for the 2D Laplacian. We introduce some new a posteriori error estimators based on the (h − h/2)-error estimation strategy. In particular, these include the approximation error for the boundary data, which allows to work with discrete boundary integral operators only. Using the concept of estimator reduction, we prove that the proposed adaptive algorithm is convergent in the sense that it drives the underlying error estimator to zero. Numerical experiments underline the reliability and efficiency of the considered adaptive mesh-refinement

A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

Global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on $R^d$ are studied. These equations constitute gradient flows for the perturbed information functionals $F[u] = 1/(2\alpha) \int | D (u^\alpha) |^2 dx + \lambda/2 \int |x|^2 u dx$ with respect to the $L^2$-Wasserstein metric. The value of $\alpha$ ranges from $\alpha=1/2$, corresponding to a simplified quantum drift diffusion model, to $\alpha=1$, corresponding to a thin film type equation.Comment: 33 pages, no figure