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.

本文介绍了一种快速无网格技术，即直接无网格局部Petrov-Galerkin（DMLPG）方法，用于识别二维抛物线逆PDE中的控制函数。DMLPG方法是一种真正的无网格技术，可以直接逼近局部弱形式。在DMLPG中，局部积分是通过低次多项式基函数而不是复杂的形状函数完成的。因此，通过对多项式进行积分来构造质量矩阵和刚性矩阵。与有限元方法（FEM）相比，此功能克服了无网格方法的主要缺点，并且与其他无网格弱形式方法相比，大大提高了计算效率。在本文中，我们应用DMLPG和MLPG方法来求解规则和不规则域上的二维抛物线逆PDE。将这些方法相互比较，并证明了DMLPG优于经典MLPG的优越性。数值结果证实了DMLPG方法解决反问题特别是系数反问题的良好效率。最后，我们将得出结论，在解决反问题时，可以将DMLPG方法视为现有无网格弱形式方法的有吸引力的替代方法。