Nonconforming FEMs for an Optimal Design Problem

Some optimal design problems in topology optimization eventually lead to a degenerate convex minimization problem $E(v):=\int_{\Omega}W(\nabla v)dx-\int_{\Omega}f\,v\,dx$ for $v \in H_0^1(\Omega)$ with possibly multiple minimizers $u$, but with a unique stress $\sigma:=DW(\nabla u)$. 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.

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