Optimal error estimates of a lowest-order Galerkin-mixed FEM for the thermoviscoelastic Joule heating equations

This paper is concerned with the optimal error estimates of a classical Galerkin-mixed finite element method (FEM) for the thermoviscoelastic Joule heating equations, which couples the temperature, the electric potential and the deformation of a thermoviscoelastic body. The method is based on a popular combination of the lowest-order Raviart-Thomas mixed approximation for the electric potential/field (ϕ,θ) and the linear Lagrange approximation for the temperature u and the deformation b. By using the temporal-spatial error splitting techniques, we prove that the method produces the optimal second-order accuracy O(h²) for u and b in the spatial direction, and the accuracy O(h) for the potential/field without any restriction on the time step size. Moreover, a simple single-step recovery method is introduced to improve the accuracy for the electric potential/field to O(h²). Numerical results are provided to confirm our theoretical analysis and show clearly that no time-step condition is needed.