A posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle problems

• Published in: Applied numerical mathematics Vol. 76; pp. 76 - 92
• Main Authors: Banz, Lothar, Stephan, Ernst P.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2014
• Subjects: Approximation; Consistency; Convergence; Elliptic obstacle problems; Errors; Estimates; hp-Adaptive IPDG-FEM; Mathematical models; Obstacles; Residual and hierarchical a posteriori error estimates; Tolerances; Elliptic obstacle problems; Residual and hierarchical a posteriori error estimates; hp-Adaptive IPDG-FEM
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2013.10.004

A variational inequality (VI) and a mixed formulation for an elliptic obstacle problem are considered. Both formulations are discretized by an hp-FE interior penalty discontinuous Galerkin (IPDG) method. In the case of the mixed method, the discrete Lagrange multiplier is a linear combination of biorthogonal basis functions. In particular, also the discrete problems are equivalent. For these formulations a residual based a posteriori error estimate and a hierarchical a posteriori error estimate are derived. For the mixed method the residual based estimate is constructed explicitly, for which the approximation error is split into a discretization error of a linear variational equality problem and additional consistency and obstacle condition terms. For the VI-method a hierarchical estimate based on an overkill solution is derived explicitly. This includes the h–h/2 and p–(p+1)-estimates. The numerical experiments show exponential convergence up to the desired tolerance.