Discontinuity-capturing operators for elastodynamics

Abstract

Finite element methods are presented which include discontinuity-capturing operators to accurately solve elastodynamics problems exhibiting sharp gradients in their solution, e.g., wave propagation. The formulation involves finite element discretization of the temporal domain as well as the usual finite element discretization of the spatial domain. Linear least-squares terms are included that enhance the stability of the space-time finite element method. Nonlinear discontinuity-capturing operators are presented that result in more accurate capturing of wave fronts in transient solutions while maintaining the high-order accuracy of the underlying linear algorithm in smooth regions. Stability of the discontinuity-capturing formulation is proved and a convergence proof is given for a particular choice of the discontinuity-capturing operator. Numerical results are presented that demonstrate the effectiveness of the proposed discontinuity-capturing operators for elastodynamics.