A alternating boundary element method for solving cauchy problems for the biharmonic equation

Abstract

In this study the Cauchy problems that can be associated to the biharmonic equation are classified into 5 distinct ill-posed formulations. The biharmonic equation is numerically discretised using the boundary element method (BEM) and the degree of ill-posedness of the inverse formulations is characterised by the condition number of the senstitivity matrix. The biharmonic Cauchy problems are mathematically algorithmised by accomodating the iterative method developed by Kozlov et al. [1] which is convergent for Cauchy problems of elliptic equations in general. The numerical implementation of the mathematical algorithm is based on an iterative application of the BEM for a sequence of mixed well-posed direct biharmonic problems. It is shown that the iterative BEM produces a convergent and accurate numerical solution with respect to increasing the number of boundary elements and the number of iterations. Care was taken to ensure that all the unspecified boundary data is accurately approximated and this was accomplished by a further application of the singular value decomposition truncated at an optimal level given by the L-curve method. Furthermore, the iterative alternating algorithm has a regularizing character and the stability of the numerical solution was shown by imposing various amounts of noise in the input data.