Semi-discrete and fully discrete mixed finite element methods for Maxwell viscoelastic model of wave propagation

Semi-discrete and fully discrete mixed finite element methods are considered for Maxwell-model-based problems of wave propagation in linear viscoelastic solid. This mixed finite element framework allows the use of a large class of existing mixed conforming finite elements for elasticity in the spatial discretization. In the fully discrete scheme, a Crank-Nicolson scheme is adopted for the approximation of the temporal derivatives of stress and velocity variables. Error estimates of the semi-discrete and fully discrete schemes, as well as an unconditional stability result for the fully discrete scheme, are derived. Numerical experiments are provided to verify the theoretical results.