The Method of Fundamental Solutions: Theory and Applications

The fundamental solutions (FS) satisfy the governing equations in a solution
domain S, and then the numerical solutions can be found from the exterior and
the interior boundary conditions on ?S. The resource nodes of FS are chosen
outside S, distinctly from the case of the boundary element method (BEM). This
method is called the method of fundamental solutions (MFS), which originated
from Kupradze in 1963. The Laplace and the Helmholtz equations are studied in
detail, and biharmonic equations and the Cauchy-Navier equation of linear
elastostatics are also discussed. Moreover, better choices of source nodes are
explored. The simplicity of numerical algorithms and high accuracy of
numerical solutions are two remarkable advantages of the MFS. However, the
ill-conditioning of the MFS is notorious, and the condition number (Cond)
grows exponentially via the number of the unknowns used. In this book, the
numerical algorithms are introduced and their characteristics are addressed.
The main efforts are made to establish the theoretical analysis in errors and
stability. The strict analysis (as well as choices of source nodes) in this
book has provided the solid theoretical basis of the MFS, to grant it to
become an effective and competent numerical method for partial differential
equations (PDE). Based on some of our works published as journal papers, this
book presents essential and important elements of the MFS. It is intended for
researchers, graduated students, university students, computational experts,
mathematicians and engineers.