The boundary element solution of the Laplace and biharmonic equations subjected to noisy boundary data

This study investigates the numerical solution of the Laplace and biharmonic equations subjected to noisy boundary data. Since both equations are linear, they are numerically discretized using the Boundary Element Method (BEM), which does not use any solution domain discretization, to reduce the problem to solving a system of linear algebraic equations for the unspeci ed boundary values. It is shown that when noisy, lower-order derivatives are prescribed on the boundary, then a direct approach, e.g. Gaussian elimination, for solving the resulting discretized system of linear equations produces an unstable, i.e. unbounded and highly oscillatory, numerical solution for the unspeci ed higher-order boundary derivatives data. In order to overcome this di culty, and produce a stable solution of the resulting system of linear equations, the singular value decomposition approach (SVD), truncated at an optimal level given by the L-curve method, is employed.