Generalized moving least squares interpolation for solution of partial differential equations
OA Version
Citation
Abstract
Methods for computing the solution of partial differential equation typically require three key ingredients, namely: (1) how to represent the simulation domain, (2) how to represent the approximate solution and (3) how to enforce the governing equation. For example, the Finite Element Method requires a mesh to satisfy conditions (1) and (2). Doing so, however, places strict requirements on the mesh that are difficult to meet in applications.
This thesis mainly concentrates on utilizing the Generalized Moving Least Squares approximation in order to fulfill requirement (2) of the three key ingredients. Thereby, we reduce the requirements on the mesh to represent the unknown functions. Generalized Moving Least Squares builds a polynomial approximation for a function by minimizing the squared residual errors at specific locations throughout the domains. In the first part, we will fulfill condition (1) by a point-cloud (particle) representation of the simulation domain and condition (3) with a finite-difference-like collocation scheme on the strong form of the Partial Differential Equations at the particle locations. We apply this scheme to solve steady-state Stokes flow. Our results indicate that the error for both velocity and pressure field exhibits a high-order convergence rate. Additionally, the performance benchmarks suggest that our parallel implementation of the method is scalable for larger systems, and thus has potential to be executed on sizeable supercomputing clusters.
In the second part, we will borrow the framework from the Finite Element Method and satisfy condition (1) with a mesh. The resulting method has compactly supported discontinuous shape functions which are generated from generalized moving least squares. These discontinuous polynomials are then applied within a Discontinuous Galerkin variational formulation with interior penalty to accomplish condition (3). Since the basic functions are separated from the shape of the underlying elements, the dependence on the mesh quality is, therefore, removed. We derive \textit{a priori} error bounds of this formulation, specifically for solving Poisson's boundary value problem and the linear elasticity problem. The numerical result demonstrates the expected convergence behavior even on poor-quality meshes. Moreover, we have found that this scheme is able to maintain much higher stability, when compared against the conventional Finite Element Methods.