Fundamental Solutions Method for Elliptic Boundary Value Problems

We consider a procedure for solving boundary value problems for elliptic homogeneous equations, known as the fundamental solutions method. We prove its applicability for some second order operators as well as for fourth order ones. The boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest. In a specific case of the Laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions. Numerical examples are given, among which are simple Signorini’s problems for harmonic and Lame operators.