Finite Element Method Based Multigrid Solvers for the Diffusion Equation and Fast Adaptive Composite-Grid Solvers for the Poisson Equation.
Stotsky, Jay Alexander.
2013
- Partial differential equations (PDEs) arise in numerous disciplines, especially in engineering and science. Many important phenomena are governed by PDEs, and their solutions often provide important theoretical and practical insight. For most PDE problems, there exist no analytical solutions so numerical methods must be employed to obtain approximations of the exact solution. The Finite Element ... read moreMethod (FEM) is a commonly used numerical method of solving PDEs. With the FEM, a partial differential equation can be converted into a set of algebraic equations which, when solved yield a good approximation of the exact solution to the PDE. Unfortunately, accurate, high resolution results require very high numbers of equations to be solved, and solving these algebraic equations becomes extremely costly at levels of accuracy that are often far too low for many applications. Multigrid methods provide a means to rapidly and efficiently solving the algebraic equations derived from the discretization of a partial differential equation. Finite Element Method based multigrid solvers will be developed to solve two different partial differential equations. The first multigrid solver examined will be used to solve an equation that arises in the study of the dispersion electron beams as they interact with matter. As a beam of electrons comes into contact with matter, the electrons are deflected as they collide with atoms in the matter. The electron density can be described by the Focker-Planck equation. One term in this equation involves a quantity that governs the direction which electrons are traveling. This direction variable can be modelled by diffusion on the surface of a sphere. An efficient FEM based multigrid scheme to solve the diffusion equation on the sphere will be developed and results shown. In the second case, a special version of the multigrid algorithm, known as the Fast Adaptive Composite Grid (FAC) method which can incorporate adaptive mesh refinement discretizations will be discussed. This extension to regular multigrid will be derived using a FEM based solver and will be used to solve the Poisson equation in one dimension and in two dimensions on the unit square.read less
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