The adaptive cross approximation (ACA) algorithm [2,3] provides a means to compute data-sparse approximants of discrete integral formulations of elliptic boundary value problems with almost linear complexity. ACA uses only few of the original entries for the approximation of the whole matrix and is therefore well-suited to speed up existing computer codes. In this article we extend the convergence proof of ACA to Galerkin discretizations. Additionally, we prove that ACA can be applied to integral formulations of systems of second-order elliptic operators without adaptation to the respective problem. The results of applying ACA to boundary integral formulations of linear elasticity are reported. Furthermore, we comment on recent implementation issues of ACA for nonsmooth boundaries.

A convolution thresholding approximation scheme for the Willmore geometric flow is constructed. It is based on an asymptotic expansion of the convolution of an indicator function with a smooth, isotropic kernel. The consistency of the method is justified when the evolving surface is smooth and embedded. Some aspects of the numerical implementation of the scheme are discussed and several numerical results are presented. Numerical experiments show that the method performs well even in the case of a non-smooth initial data.

This is the post-print version of the article. The official published version can be accessed from the links below - Copyright @ 2013 Springer-VerlagA numerical implementation of the direct boundary-domain integral and integro-differential equations, BDIDEs, for treatment of the Dirichlet problem for a scalar elliptic PDE with variable coefficient in a three-dimensional domain is discussed. The mesh-based discretisation of the BDIEs with tetrahedron domain elements in conjunction with collocation method leads to a system of linear algebraic equations (discretised BDIE). The involved fully populated matrices are approximated by means of the H-Matrix/adaptive cross approximation technique. Convergence of the method is investigated.This study is partially supported by the EPSRC grant EP/H020497/1:"Mathematical Analysis of Localised-Boundary-Domain Integral Equations for Variable-Coefficients\ud Boundary Value Problems"

SUMMARYThe purpose of this paper is to present an adaptive finite element-boundary element method (FEM-BEM) coupling method that is valid for both two-and three-dimensional elasto-plastic analyses. The method takes care of the evolution of the elastic and plastic regions. It eliminates the cumbersome of a trial and error process in the identification of the FEM and BEM sub-domains in the standard FEM-BEM coupling approaches. The method estimates the FEM and BEM sub-domains and automatically generates/adapts the FEM and BEM meshes/sub-domains, according to the state of computation. The results for two-and three-dimensional applications in elasto-plasticity show the practicality and the efficiency of the adaptive FEM-BEM coupling method.

The phenomenon of physical contact between red blood cells and artificial surfaces is considered. A fully three-dimensional mathematical model of a bilayer membrane in contact with an artificial surface is presented. Numerical results for the different geometries and adhesion intensities are found to be in agreement with experimentally observed geometries obtained by means of digital holographic microscopy.