In this dissertation, we study numerical algorithms for time dependent problems in continuum mechanics using mixed finite element methods. We are particularly interested in linear elastodynamics and the Kelvin--Voigt, Maxwell, and generalized Zener models in linear viscoelasticity. We use mixed finite elements for elasticity with weak symmetry of stress, and show the a priori error analysis. A main contribution of our analysis is proving existence of a new elliptic projection map, called a weakly symmetric elliptic projection. In our analysis we prove that a priori error estimates for elastodynamics and viscoelasticity problems are as good as that of stationary elasticity problems. We present numerical results supporting our error analysis. We also present some basic numerical simulations which are more involved in physical situations.
University of Minnesota Ph.D. dissertation. July 2012. Major: Mathematics. Advisor:Douglas N. Arnold. 1 computer file (PDF); vii, 142 pages.
Mixed methods with weak symmetry for time dependent problems of elasticity and viscoelasticity..
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