Nonconforming FEMs for an Optimal Design Problem

Some optimal design problems in topology optimization eventually lead to a degenerate convex minimization problem E(v) := ∫ Ω W (∇v)dx − ∫ Ω f v dx for v ∈ H1 0 (Ω) with possibly multiple minimizers u, but with a unique stress σ := DW (∇u). This paper proposes the discrete Raviart–Thomas mixed finite element method (dRT-MFEM) and establishes its equivalence with the Crouzeix–Raviart nonconforming finite element method. The convergence analysis combines the a priori convergence rate of the conforming FEM with the efficient a posterior error control of MFEM. Numerical experiments provide empirical evidence that the proposed dRT-MFEM overcomes the reliability-efficiency gap for the first time.