Mixed finite element methods for nonlinear equations: a priori and a posteriori error estimates

Abstract

A priori error estimation provides information about the asymptotic behavior of the approximate solution and information on convergence rates of the problem. Contrarily, a posteriori error estimation derives the estimation of the exact error by employing the approximate solution and provides a practical accurate error estimation. Additionally, a posteriori error estimates can be used to steer adaptive schemes, that is to decide the refinement processes, namely local mesh refinement or local order refinement schemes. Adaptive schemes of finite element methods for numerical solutions of partial differential equations are considered standard tools in science and engineering to achieve better accuracy with minimum degrees of freedom.

In this thesis, we focus on a posteriori error estimations of mixed finite element methods for nonlinear time dependent partial differential equations. Mixed finite element methods are methods which are based on mixed formulations of the problem. In a mixed formulation, the derivative of the solution is introduced as a separate dependent variable in a different finite element space than the solution itself. We implement the $H^1$-Galerkin mixed finite element method (H1MFEM) to approximate the solution and its derivative. Two nonlinear time dependent partial differential equations are considered in this thesis, namely the Benjamin-Bona-Mahony (BBM) equation and Burgers equation. Our a posteriori error estimations are based on implicit schemes of a posteriori error estimations, where the error estimators are locally computed on each element. We propose a posteriori error estimates by using the approximate solution produced by H1MFEM and use the a posteriori error estimates to compute the local error estimators, respectively for the BBM and Burgers equations. Then, we prove that the introduced a posteriori error estimates are accurate and efficient estimations of the exact errors.

The last part of this study is on numerical studies of adaptive mesh refinement schemes for the two equations mentioned above. By implementing the introduced a posteriori error estimates, we propose adaptive mesh refinement schemes of H1MFEM for both equations.