MLPG approximation to the p-Laplace problem

Abstract

Meshless local Petrov-Galerkin (MLPG) method is discussed for solving 2D, nonlinear, elliptic p-Laplace or p-harmonic equation in this article. The problem is transferred to corresponding local boundary integral equation (LBIE) using Divergence theorem. The analyzed domain is divided into small circular sub-domains to which the LBIE is applied. To approximate the unknown physical quantities, nodal points spread over the analyzed domain and MLS approximation, are utilized. The method is a meshless method, since it does not require any background interpolation and integration cells and it dose not depend on geometry of domain. The proposed scheme is simple and computationally attractive. Applications are demonstrated through illustrative examples.

The finite point method for the p-Laplace equation

“…The examples are chosen for comparison with the results of [6,22]. In both cases we have used the quartic spline weight function:…”

“…As mentioned in [4][5][6], if we consider the isothermal Hele-Shaw flows, which physically arise whenever the fluid viscosity does not depend on temperature, the problems are related to solve the following 2D, non-linear, elliptic equation div(|∇u| γ ∇u) = 0, (1.2) which is [22] a p-Laplace equation of index p = γ + 2. The solution yields the pressure distribution u(x, y) in the filled region of the mould with boundary .…”

“…The p-Laplace equation, p u = 0, plays an important role in the modeling of many phenomena in several areas such as glaciology, non-Newtonian rheology or edge-preserving image deblurring [22].…”

“…In [6] the authors were linearized the equation and applied a meshfree technique based on RBF approximation introduced by G. Fasshauer which allows to solve it in the framework of Kansa method. In [22] the meshfree local Petrov Galerkin (MLPG) technique is developed for solving the p-Laplace equation.…”

“…We will refer to (1.3) and (1.4) as Dirichlet-injection and Neumann-injection boundary conditions, respectively. The numerical simulation of the Hele-Shaw flow requires some methods [22] for solving equation (1.2) at every time step with some techniques to advance the front to its new position, until the mould domain has been completely filled [4,5]. Once the pressure is known by solving the above equations, the velocity at each point in the moving front is computed as…”