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Poullikkas, A.

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The method of fundamental solutions for the solution of elliptic boundary value problems

We investigate the use of the Method of Fundamental Solutions (MFS) for the numerical solution
of elliptic problems arising in engineering. In particular, we examine harmonic and biharmonic
problems with boundary singularities, certain steady-state free boundary flow problems and
inhomogeneous problems. The MFS can be viewed as an indirect boundary method with an
auxiliary boundary. The solution is approximated by a linear combination of fundamental
solutions of the governing equation which are expressed in terms of sources located outside the
domain of the problem. The unknown coefficients in the linear combination of fundamental
solutions and the location of the sources are determined so that the boundary conditions are
satisfied in a least squares sense. The MFS shares the same advantages of the boundary methods
over domain discretisation methods. Moreover, it is relatively easy to implement, it is adaptive
in the sense that it takes into account sharp changes in the solution and/or in the geometry of
the domain and it can easily incorporate complicated boundary conditions. [Continues.

The method of fundamental solutions for the solution of elliptic boundary value problems

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