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

摘要

A priori error estimation provides informationabout the asymptotic behaviorof the approximate solution andinformationon convergence rates of the problem. Contrarily,a posteriori error estimation derives the estimation of the exact errorby employing the approximate solution and providesa 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 orlocal order refinement schemes. Adaptive schemes offinite element methods for numerical solutions of partial differential equationsare considered standard toolsin science and engineering to achieve better accuracywith minimum degrees of freedom.In this thesis, we focus on a posteriori error estimationsof mixed finite element methods for nonlineartime dependent partial differential equations.Mixed finite element methods are methods which are based on mixedformulations of the problem. In a mixed formulation,the derivative of the solution is introduced as aseparate 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 errorestimations, where the error estimators are locally computed on each element.We propose a posteriori error estimates by using the approximate solution produced by H1MFEMand 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 meshrefinement 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.