A local discontinuous Galerkin method for the fourth-order Euler–Lagrange partial differential equation for beams
The discontinuous Galerkin (DG) method is a class of finite element methods which use completely discontinuous functions as the numerical solutions and test functions. DG method has been under rapid development in recent years as one of the powerful numerical methods to solve various partial differential equations (PDEs), especially convection dominated PDEs in science and engineering. Stable and convergent discontinuous Galerkin methods have been designed in recent years for various types of PDEs, including hyperbolic, parabolic and elliptic equations. DG methods have found rapid applications recently, in such diverse areas as aeroacoustics, electromagnetism, gas dynamics, granular flows, meteorology, oceanography, semiconductor device simulation, turbulent flow, weather forecasting, among many others.
The local discontinuous Galerkin (LDG) finite element method is an extension of the discontinuous Galerkin (DG) method aimed at solving ordinary and partial differential equations containing higher than first-order spatial derivatives. In this project, we design a new LDG method for solving the classical fourth-order Euler-Bernoulli partial differential equation. We propose to investigate superconvergence properties of LDG solutions determine their dependence on the numerical flux function, stabilization strategy, mesh structure, etc. We will perform a numerical and theoretical study of the existence of superconvergence points. Finally, we will use the superconvergence results to construct simple, efficient, and asymptotically correct a posteriori error estimates by solving local problems with no boundary conditions on each element. A posteriori error estimates are needed to guide adaptive enrichment and to provide a measure of solution accuracy for any numerical method. Numerical experiments will be presented to validate the theoretical results.
MATH 3300 and MATH 4330 are required. You will be using Matlab.