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Energy norm a posteriori error estimation for discontinuous Galerkin methods. (English) Zbl 1042.65083
The authors present a residual-based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of dicontinuous Galerkin (dG) approximations of elliptic problems. The estimate is of optimal order and is valid for a general family of dG methods including the classical symmetric method of J. Nitsche [Abh. Math. Semin. Univ. Hamburg 36, 9–15 (1971; Zbl 0229.65079)], the recent nonsymmetric method proposed by J. T. Oden, I. Babuška and C. E. Baumann [J. Comput. Phys. 146, 491–519 (1998; Zbl 0926.65109)], and stabilized versions thereof. The theory is developed for an elliptic problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed. The illustrating numerical results indicate that the effectivity index is not severely effected by the fineness of the mesh.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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