Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems

¹ Center for Applied Mathematics, Purdue University, West Lafayette, IN 47907-1395, USA. Supported in part by the NSF and the ONR. douglas@math.purdue.edu.
² Center for Applied Mathematics, Purdue University, West Lafayette, IN 47907-1395, USA, and CONICET, Observatorio Astronomico, Universidad Nacional de La Plata, La Plata 1900, Argentina.
³ Department of Mathematics, Seoul National University, Seoul 151-742, Korea. Supported in part by KOSEF-GARC and BSRI-MOE-97.
⁴ Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204-1099, USA.

Received: 3 August 1998

Abstract

Low-order nonconforming Galerkin methods will be analyzed for second-order elliptic equations subjected to Robin, Dirichlet, or Neumann boundary conditions. Both simplicial and rectangular elements will be considered in two and three dimensions. The simplicial elements will be based on P₁, as for conforming elements; however, it is necessary to introduce new elements in the rectangular case. Optimal order error estimates are demonstrated in all cases with respect to a broken norm in H¹(Ω) and in the Neumann and Robin cases in L²(Ω).