Superconvergence analysis for patch-recovered finite element approximations of topological derivatives

Abstract The topological derivative for a general shape functional in shape and topology optimization typically involves the gradients of partial differential equation solutions to the physical field(s). Under continuous piecewise linear finite element discretization in literature the discrete topological derivative is discontinuous and exhibits low accuracy of only first-order convergence with respect to the mesh size. To recover continuity and improve accuracy we utilize the superconvergence patch recovery techniques, e.g., weighted averaging and local least-squares fitting to develop a novel topological derivative formula that is continuous and exhibits superlinear convergence. Superconvergence properties for the novel topological derivative are investigated for shape and topology optimization of both elliptic problem (in $L^{2}$ and $L^\infty $-norm) and elasticity system (in $L^{2}$-norm). Numerical examples on triangular and rectangular grids have been verified to show convergence of the proposed formulas.