Semi-discrete Weak Galerkin Finite Element Method of Viscoelastic Wave Propagation

This paper primarily proposes a weak Galerkin finite element of arbitrary order for the wave propagation equation of the Maxwell viscoelasticity model. The quasi-static viscoelastic equations of the Maxwell model serve as the foundation for this approach. By introducing the velocity variable in the model problem, the weak variational form of the velocity-stress model is established. The spatial variables are then discretized using the weak Galerkin finite element method, resulting in a semi-discrete format for the velocity-stress model. In the discretization process of the weak Galerkin finite element method, piecewise polynomials of degree k (k≥1) and k+1 are used to approximate the stress and velocity within each element, and piecewise polynomials of degree k are employed to approximate the trace of the velocity on the element boundaries. By analyzing and proving the method, this paper demonstrates the stability of the semi-discrete format's solution using Gronwall's inequality, and finally establishes the optimal error estimation with respect to the spatial mesh size for this method. The research findings of this paper indicate that by introducing the velocity variable and utilizing the weak Galerkin finite element method, the wave propagation equation of the viscoelasticity based on the Maxwell model can be effectively solved.