Finite element approximations of nonlinear elastic waves

Makridakis, Charalambos G.

doi:10.1090/S0025-5718-1993-1195426-X

Finite element approximations of nonlinear elastic waves
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Math. Comp. 61 (1993), 569-594 Request permission

Abstract:

In this paper we study finite element methods for a class of problems of nonlinear elastodynamics. We discretize the equations in space using Galerkin methods. For the temporal discretization, the construction of our schemes is based on rational approximations of $\cos x$ and ${e^x}$. We analyze semidiscrete as well as second- and fourth-order accurate in time fully discrete methods for the approximation of the solution of the problem and prove optimal-order ${L^2}$ error estimates. For some schemes a Taylor-type technique is used so that only linear systems of equations need be solved at each time step. In the proofs we need various estimates for a nonlinear elliptic projection, the proofs of which are also established in the paper.

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