Direct meshless local Petrov–Galerkin method for elliptic interface problems with applications in electrostatic and elastostatic

In recent years, there have been extensive efforts to find the numerical methods for solving problems with interface. The main idea of this work is to introduce an efficient truly meshless method based on the weak form for interface problems. The proposed method combines the direct meshless local Petrov–Galerkin method with the visibility criterion technique to solve the interface problems. It is well-known in the classical meshless local Petrov–Galerkin method, the numerical integration of local weak form based on the moving least squares shape function is computationally expensive. The direct meshless local Petrov–Galerkin method is a newly developed modification of the meshless local Petrov–Galerkin method that any linear functional of moving least squares approximation will be only done on its basis functions. It is done by using a generalized moving least squares approximation, when approximating a test functional in terms of nodes without employing shape functions. The direct meshless local Petrov–Galerkin method can be a very attractive scheme for computer modeling and simulation of problems in engineering and sciences, as it significantly uses less computational time in comparison with the classical meshless local Petrov–Galerkin method. To create the appropriate generalized moving least squares approximation in the vicinity of an interface, we choose the visibility criterion technique that modifies the support of the weight (or kernel) function. This technique, by truncating the support of the weight function, ignores the nodes on the other side of the interface and leads to a simple and efficient computational procedure for the solution of closed interface problems. In the proposed method, the essential boundary conditions and the jump conditions are directly imposed by substituting the corresponding terms in the system of equations. Also, the Heaviside step function is applied as the test function in the weak form on the local subdomains. Some numerical tests are given including weak and strong discontinuities in the Poisson interface problem. To demonstrate the application of these problems, linearized Poisson–Boltzmann and linear elasticity problems with two phases are studied. The proposed method is compared with analytical solution and the meshless local Petrov–Galerkin method on several test problems taken from the literature. The numerical results confirm the effectiveness of the proposed method for the interface problems and also provide significant savings in computational time rather than the classical meshless local Petrov–Galerkin method.