Meshless solution of a diffusion equation with parameter optimization and error analysis

Derivation and implementation of a numerical solution of a time-dependent diffusion equation is given in detail, based on the meshless local Petrov–Galerkin method (MLPG). A simple method is proposed that ensures a constant number of support nodes for each point. Numerical integrations are carried out over local square domains. The implicit Crank–Nicolson scheme is used for time discretization. A detailed convergence study was performed experimentally to optimize the number of support nodes, quadrature domain size and other parameters. The accuracy of the MLPG solution is compared with that of standard methods on a unit square and on an irregularly shaped test domain. As expected, the finite difference method on a regular mesh is incompetitive on irregularly shaped domains. MLPG is significantly more accurate when using moving least square shape functions of degree two than with degree one. It is comparable to the finite element method of degree two in the H1H1 error norm and about two times less accurate in the L2L2 error norm.