hp-adaptive Interior Penalty FEM for Elliptic Obstacle Problems

Abstract: 

Firstly, from [1] we consider a mixed formulation for an elliptic obstacle problem for a 2nd order operator and present an hp-FE interior penalty discontinous Galerkin (IPDG) method. The primal variable is approximated by a linear combination of Gauss-Lobatto-Lagrange(GLL)-basis functions, whereas the discrete Lagrangian multiplier is a linear combination of biorthogonal basis functions. A residual based a posteriori error estimate is derived. For its construction the approximation error is split into a discretization error of a linear variational equality problem and additional consistency and obstacle condition terms. Secondly, an hp-adaptive C0-interior penalty method for the bi-Laplace obstacle problem is presented from [2]. Again we take a mixed formulation using GLL-basis functions for the primal variable and biorthogonal basis functions for the Lagrangian multiplier and present also a residual a posteriori error estimate. For both cases (2nd and 4th order obstacle problems) our numerical experiments clearly demonstrate the superior convergence of the hp-adaptive schemes compared with uniform and h-adaptive schemes.

1. L.Banz, E. P. Stephan, A posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle problems, Applied Numerical Mathematics 76,(2014) 76-92.
2. L. Banz, B. P. Lamichhane, E. P. Stephan, An hp-adaptive C0-interior penalty method for the obstacle problem of clamped Kirchhoff plates, preprint (2015).