DMLPG method for specifying a control function in two-dimensional parabolic inverse PDEs

Abstract

This article describes a fast meshless technique, the direct meshless local Petrov–Galerkin (DMLPG) method, for identifying a control function in two-dimensional parabolic inverse PDEs. The DMLPG method is a truly meshless technique that directly approximates the local weak forms. In DMLPG, the local integrations are done over the low-degree polynomial basis functions instead of the complicated shape functions. Thus, the mass and stiff matrices are constructed by integration against polynomials. This feature overcomes the main drawback of meshless methods in comparison with the finite element methods (FEM) and significantly increases the computational efficiency in comparison with the other meshless weak form methods. In this paper, we apply the DMLPG and MLPG methods for solving two-dimensional parabolic inverse PDEs on regular and irregular domains. These methods are compared with each other and the superiority of the DMLPG over the classical MLPG is demonstrated. The numerical results confirm the good efficiency of the DMLPG method for solving inverse problems especially coefficient inverse problems. Finally, we will conclude that the DMLPG method can be considered as an attractive alternative to existing meshless weak form methods in solving inverse problems.