Mixed Galerkin finite element methods for fourth order differential equations

Abstract

The work presented in this thesis is on numerical schemes, optimal order a priori error estimates and computational experiments for fourth order differential equations using mixed Galerkin finite element methods. Two types of ordinary differential equations and two types of nonlinear time-dependent partial differential equations of fourth order in single space variable are considered. A quadrature based mixed Petrov-Galerkin finite element method is applied to a special type of fourth order linear ordinary differential equation in divergence form. The integrals are then replaced by Gauss quadrature rule in the formulation itself. Optimal order a priori error estimates are obtained without any restriction on the mesh. The same method is then applied to a general fourth order linear ordinary differential equation and optimal order a priori error estimates are obtained without any restriction on the mesh. These error estimates are validated by a numerical example. An H1-Galerkin mixed finite element method is applied to the extended Fisher-Kolmogorov equation, a nonlinear time dependent fourth order partial differential equation, employing a splitting technique. This method may also be considered as a Petrov-Galerkin method with cubic spline space as trial space and piecewise linear space as test space, since second derivative of a cubic spline is a linear spline. A fully discrete scheme is also developed and optimal order a priori error estimates are obtained. The results are validated with numerical examples. A similar method is applied to the Kuramoto-Sivashinsky equation which is also a nonlinear time dependent fourth order partial differential equation. By employing a splitting technique, optimal order a priori error estimates are obtained without any restriction on the mesh. A fully discrete scheme is also discussed and optimal order a priori error estimates are obtained. The results are validated with numerical examples. newline newline newline