Robust a posteriori error estimation for nonconforming finite element approximation. (English) Zbl 1085.65102
The equilibrated residual method provides a posteriori estimates without generic constants in the main term and can also be found with the name hypercircle method. The essential idea is a comparison of a primal and a dual variational problem and may be traced back to Prager and Synge [1949]. The method is extended here to the nonconforming element of M. Crouzeix and P.-A. Raviart [Rev. Franc. Automat. Inform. Rech. Operat. 7(1973), R-3, 33–76 (1974; Zbl 0302.65087)]. Roughly speaking the error is split into a conforming and a nonconforming part. To this end a Helmholtz decomposition is applied to the nonconforming gradient. When there are jumps in the coefficients, the nonmonotonicity of the jumps on paths around a node gives rise a multiplicative factor in the error bound.
Reviewer: Dietrich Braess (Bochum)
MSC:
65N15 | Error bounds for boundary value problems involving PDEs |
65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
35J25 | Boundary value problems for second-order elliptic equations |