Finite Element Approximations of High Order Partial Differential Equations
Open Access
- Author:
- Zheng, Bin
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 06, 2008
- Committee Members:
- Jinchao Xu, Committee Chair/Co-Chair
Chun Liu, Committee Member
Eric M Mockensturm, Committee Member
Victor Nistor, Committee Member
Ludmil Tomov Zikatanov, Committee Member - Keywords:
- finite element methods
high order partial differential equations
magnetohydrodynamics - Abstract:
- Developing accurate and efficient numerical approximations of solutions of high order partial differential equations (PDEs) is a challenging research topic. In this dissertation, we study finite element approximations of high order PDEs that arise in many physics and engineering applications. A common method of solving a high order PDE is to split it into a system of lower order equations. By carefully studying the biharmonic equation with different types of boundary conditions, we are able to justify the fact that the lower order system of equations and the original problem may have different solutions. Our analysis shows that direct discretizations are much better suited for the numerical solution of high order problems. We construct two finite elements to directly discretize high order equations arising from magnetohydrodynamics (MHD) models. These elements provide nonconforming approximations for which the number of degrees of freedom is much smaller than that of a conforming method. The inter-element continuity is only imposed along the tangential directions which is appropriate for the approximation of the magnetic field. A detailed construction of basis functions for the new elements is given, and we also prove that these finite element approximations converge for a model problem containing both second order and fourth order terms. Another important property of high order PDEs that model physical phenomena in material sciences, fluid mechanics and plasma physics is that they often involve different time and spatial scales. The solutions exhibit sharp interfaces, such as shocks, current sheets and other singularities. Adaptive mesh refinement techniques are therefore crucial for reliable numerical computations of high order problems. We develop a post-processing derivative recovery scheme and a posteriori error estimates that can be used in local adaptive mesh refinement. A nice feature of the scheme is that it is independent of the PDE and a single implementation can be used to solve many different problems.