An $hp$-adaptive discontinuous Galerkin discretization of a static anti-plane shear crack model

We propose an $hp$-adaptive discontinuous Galerkin finite element method (DGFEM) to approximate the solution of a static crack boundary value problem. The mathematical model describes the behavior of a geometrically linear strain-limiting elastic body. The compatibility condition for the physical variables, along with a specific algebraically nonlinear constitutive relationship, leads to a second-order quasi-linear elliptic boundary value problem. We demonstrate the existence of a unique discrete solution using Ritz representation theory across the entire range of modeling parameters. Additionally, we derive a priori error estimates for the DGFEM, which are computable and, importantly, expressed in terms of natural energy and $L^2$-norms. Numerical examples showcase the performance of the proposed method in the context of a manufactured solution and a non-convex domain containing an edge crack.