Boundary (topology)

Boundary value problem

Applied mathematics

Convergence (economics)

Stability (learning theory)

Helmholtz equation

Economics

Computer science

Mathematical optimization

Dentistry

Neumann boundary condition

Penalty method

Thermodynamics

Physics

Finite element method

Calculus (dental)

Singular boundary method

Mathematics

Method of fundamental solutions

Mathematical analysis

Boundary element method

Uniqueness

Medicine

Economic growth

Machine learning

Engineering Analysis with Boundary Elements

In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heuristic numerical method. It is truly meshless. Its concept and numerical implementation are simple. It has the flexibility of using various forms of fundamental solutions, singular, hypersingular, or nonsingular, and mixing with general solutions and particular solutions, for different purposes. The collocation matrix, however, is not guaranteed to be invertible. There are other issues. For example, in using the logarithmic fundamental solution, a degenerate scale can exist, causing the nonuniqueness of solution. For metaharmonic operators, such as the Helmholtz equation, the zeros of the fundamental solutions can create spurious resonance frequencies. These and other issues are discussed, and remedies are offered. The traditional error analysis for the MFS shows that when the boundary condition is prescribed by a harmonic function, the convergence is exponential either by increasing the number of terms in the approximation, or by increasing the radius of the fictitious boundary. In practical problems, however, a harmonic boundary condition hardly exists. Numerical experiments show that the sources need to stay close to the boundary, and there is an optimal distance. Based on the maximum principle, a posteriori error can be monitored on the boundary to seek the optimal fictitious boundary location. Other topics discussed include the origin of the MFS, the equivalence between the MFS and the Trefftz collocation method, effective condition number, nonsingular MFS, and solving ill-posed and inverse problems.

An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability