An expanded mixed finite element approach via a dual-dual formulation and the minimum residual method (Q5946111)

Consider a simply connected bounded domain \(\Omega \) in the euclidean plane, with Lipschitz continuous boundary \(\Gamma \). The authors consider the nonhomogeneous Dirichlet problem, namely, find \(u\in H^1(\Omega)\) such that NEWLINE\[NEWLINE -\text{div}(\kappa \nabla u) = f ,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu=g\text{ on} \Gamma ,NEWLINE\]NEWLINE where \(f\in L^2(\Omega)\), \(g\in H^{-1/2}(\Gamma)\) and \(\kappa \) is a matrix-valued continuous function, symmetric and uniformly positive-definite. Following \textit{Z. Chen} [RAIRO, Modélisation Math. Anal. Numér. 32, No.~4, 479-499 and 501-520 (1998; Zbl 0910.65079 and Zbl 0910.65080)], the problem is expanded into NEWLINE\[NEWLINE u=g \text{ on } \Gamma , \quad \theta = \nabla u \text{ in } \Omega, NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sigma = \kappa \theta \text{ in } \Omega \text{ and }\text{div} \sigma = -f \text{ in } \Omega. NEWLINE\]NEWLINE By considering appropriate spaces, it is shown that the variational formulation of the expanded problem can be considered as a dual- dual mixed formulation, for which existence and uniqueness is proved. Then, for Raviart-Thomas finite element spaces, it is also shown that the Galerkin scheme for the expanded approach has a unique solution and that the approximation error is \(O(h)\), where \(h\) is the least upper bound of the diameters of the triangles involved. Since the stiffness matrix in the discretization of the expanded problem is symmetric but indefinite, the minimum residual method (MINRES) is chosen to find a numerical solution. For an example in the unit square, the reported results show that preconditioned MINRES is much more efficient than unpreconditioned MINRES.