WEB-Spline-based mesh-free finite element approximation forp-Laplacian

Abstract

In this paper, we discuss the approximation of p-Laplace problem using WEB-Spline based mesh free finite elements. Along with usual weak formulation, we also consider the mixed formulation of the pLaplace problem. We give existence, uniqueness results for both continuous and discrete problems. We also provide a priori error estimates for both the formulations.

An efficient meshless radial point collocation method for nonlinear p-Laplacian equation

“…In this section, four numerical examples are solved to demonstrate the efficiency and accuracy of the present SMRPI method. These examples were solved by other methods such as EFG method [16], IEFG method [17], FVM [14], the DG method [8] and the MFEM [7,23]. Since the best results of these methods are obtained using the EFG method, the comparison in this paper has been done between our results with those obtained in [16] and we omit any other attempts.…”

“…. , n, being the distance between nodes in the support domain, R k,i = R i (r k ) are the RBFs in (7). We added the following m equations in (5) to make a square matrix:…”

“…Other methods used to solve different types of this problems include the finite volume method [14], the finite element method [6], and the finite difference method [24]. Many meshless methods, as a special class of spectral methods, have been used for solving the p-Laplacian equation such as the Petrov-Galerkin method [21], the radial basis functions (RBFs)-based [4] and the WEB-spline-based mesh-free method [7]. On the contrary of easy implementation of these methods, the challenge of these methods is their low accuracy and high cost of computing in discretization.…”

A Multi-Scale DNN Algorithm for Nonlinear Elliptic Equations with Multiple Scales

“…In the last decades, much effort has been made for the numerical solution of the (1.1). In particular, for p-Laplacian equation with κ(x) = 1, Some degrees of effectiveness can be achieved by mesh based methods such as finite element method (FEM) [2,3,25], finite difference method (FDM) [31], discontinuous Galerkin method [12], and meshless methods [8,29] etc.. In addition to those discretization methods, iterative methods such as preconditioned steepest descent, quasi-Newton or Newton method are employed to deal with the nonlinearity.…”