Finite element methods for variational problems based on nonconforming dual mixed discretisations (Q2752287)

An elliptic boundary value problem is considered which can be solved by minimizing its associated energy integral. Alternatively, a complementary energy principle may be invoked. The primary energy integral and its complementary energy are related by an inequality. A new hybrid mixed discretisation scheme for the dual formulation is presented. Mixed finite elements of Raviart-Thomas-type are used. A novel approach to the definition of Lagrange multipliers for the continuity constraints is given. A conforming primal approximation is obtained by post-processing the multipliers, while the dual approximation need no longer be conforming.NEWLINENEWLINENEWLINEThe new scheme complements the established equivalence between primal and dual methods. Convergence is proved, rates of convergences are given. Numerical experiments are reported in the last section, where the partial differential equation NEWLINE\[NEWLINE{\partial^2u\over\partial x^2}+ {\partial^2u\over\partial x \partial y}+ 2{\partial^2u\over\partial y^2}= {17\over 8} \pi^2\cos\Biggl({3\pi x\over 2}\Biggr)\sin(\pi y)+{3\over 4}\pi^2\sin\Biggl({3\pi x\over 2}\Biggr)\cos(\pi y)- 6NEWLINE\]NEWLINE with Dirichlet boundary condition on \((-1, 1)\times (-1, 1)\) is solved numerically.