Computationally efficient r−adaptive graded meshes over non-convex domains

This study explores the use of r-adaptive mesh refinement strategies for elliptic partial differential equations (PDEs) posed on non-convex domains. We introduce an r-adaptive strategy based on a simplified optimal transport method to create a graded mesh, distributing the interpolation error evenly, considering the solution's local asymptotic behaviour. The grading ensures good mesh compression and regularity, regardless of dimension or location. We showcase our approach by studying discontinuous Galerkin (dG) finite element approximations. We utilise a posteriori error estimates for the dG method on general meshes, showing equidistribution across our graded mesh. Numerical tests on ‘L-shaped’ and ‘crack’ domains confirm that our method achieves optimal convergence rates.