On the Convergence of H1-Galerkin Mixed Finite Element Method for Parabolic Problems
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2008
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Abstract
The purpose of the present work is to study the convergence of H1-Galerkin mixed Dnite element method for the linear parabolic partial diDerential equations. The emphasis is on the theoretical aspects of such methods. An attempt has been made in this thesis to study the error analysis for the semidiscrete and fully discrete schemes with lesser regularity assumptions on the initial data. More precisely, for homogeneous parabolic problem an energy technique is used to obtain error estimates of order O(h2tD1=2) with positive time in the L2-norm for both the solution and the Dux when the given initial data is in H2(D)\H1 0 (D). Further, a parabolic duality argument is used to obtain optimal order error estimates of order O(h2tD1) for both the solution and its Dux when the given initial function is only in H1 0 (D). Analogous results are shown to hold for two dimensional parabolic problems. Since the smoothing property of the exact solution plays a signiDcant role in the study of error analysis in the semidiscrete case we, therefore, Drst investigate the smoothing property of the exact solution of this problem using energy arguments. Based on backward Euler method, a fully discrete scheme is analyzed for one-dimensional homogeneous parabolic problems and almost optimal order error bounds are established. Optimal order error estimate is the best that one can get between the exact solution and its numerical approximation when measured globally on the computational domain. But, there are places (points or lines) in the computational domain where the approximate solution is more closer to the exact solution than what is predicted by the global error estimates. It would be advantageous to make use of those points or lines in the modelling process. Therefore, we study superconvergence phenomenon for the semidiscrete H1-Galerkin mixed Dnite element method for parabolic problems. A new approximate solution for the Dux with superconvergence of order O(hk+3) is realized via a postprocessing technique, where k D 1 is the order of the approximating polynomials employed in the Raviart-Thomas element. A priori error estimates can give asymptotic rates of convergence as the mesh parameter goes to zero, but often can not provide much practical information about the actual errors encountered on a given mesh. The question of quantifying the error brings attention to a posteriori estimates. A posteriori error estimators are computable quantities which bound the errors or approximate the errors by computed numerical solution and input data of the problem. Further, to guarantee a good convergence behavior of the discrete solution, one needs to apply a reDnement algorithm based on a posteriori error estimates. Therefore, we study a posteriori error analysis for the semidiscrete and fully discrete H1-Galerkin mixed Dnite element method for parabolic problems. The estimators are derived based on a residual approach...
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Supervisor: Rajen Kr. Sinha
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MATHEMATICS