A Celatus method for the efficient solution of linear elastic problems with re-entrant corner singularities

This paper presents an enrichment method for the case of re-entrant corner singularities in linear elastic problems but with no enrichment of the finite element solution space. Instead an a posteriori error estimator, with gradient descent, is used to determine the approximate solution of the coefficients for the enrichment functions about each re-entrant corner. The method then approximately removes the singularities from the problem, increasing its regularity. As a result is that exponential convergence of the error can be achieved with uniform refinement in polynomial order. Almost no improvement in the error is expected or observed if uniform refinement in p is used. The approach is termed the Celatus method as the singularities are hidden from view . As problems are made regular it is shown that exponential convergence rates are observed when the Celatus method is combined with h p -adaptivity ( h p -Celatus), requiring far fewer degrees of freedom compared, by orders of magnitude, to traditional finite element analysis with h p -adaptivity. Furthermore each term for the enrichment functions for every re-entrant corner can be evaluated independently. Therefore the method can be implemented in an inherently parallel way. The proposed approach offers an ≈ 10 times reduction in computation time for the same accuracy compared to standard finite element analysis with h p -adaptivity. Additionally, since the solution space is not enriched and is always polynomial, the issue of having near singular matrices does not exist for the Celatus method. The discontinuous Galerkin finite element method is used here, but all equations and methodology are equally applicable to the continuous Galerkin method. • Enrichment of solution achieved without enriching the finite element solution space. • Each corner’s enrichment terms are evaluated independently for parallel execution. • Requires orders of magnitude fewer degrees of freedom than traditional hp-adaptive FEM. • Singularities are removed, converting the problem from non-smooth to smooth, providing convergent results with increasing polynomial order only. • Method avoids issues with near-singular matrices common in enriched methods.