A priori and a posteriori analyses of the symmetric interior penalty Galerkin method for a quasi-static fractional-order viscoelasticity model

Power-law type stress relaxation leads to a fractional-order viscoelasticity model. We consider a quasi-static viscoelasticity model of a fractional order which is mathematically expressed as a Volterra integral of the second kind with a weakly singular kernel. By employing the symmetric interior penalty Galerkin (SIPG) method and linear interpolation in time, we derive a fully discrete problem. To mitigate temporal discretization errors from a physical perspective in the quasi-static state, we introduce an auxiliary discrete velocity using the Crank–Nicolson approximation. We present a priori stability and error analyses for both semi-discrete and fully discrete schemes. In addition, we provide a residual-based energy norm a posteriori error estimation. Finally, we conduct numerical experiments to validate the error estimate theorems and demonstrate their applicability using real material data.