AN INTERIOR PENALTY DISCONTINUOUS GALERKIN METHOD FOR A CLASS OF MONOTONE QUASILINEAR ELLIPTIC PROBLEMS

作者：Fick Peter W^*

来源：International Journal of Numerical Analysis and Modeling, 2015, 12(4): 750-777.

• hp-discontinuous Galerkin methods
• interior penalty methods
• second-order quasilinear elliptic problems

A family of interior penalty hp-discontinuous Galerkin methods is developed and analyzed for the numerical solution of the quasilinear elliptic equation - del . (A(del u)del u) = f posed on the open bounded domain Omega subset of R-d, d >= 2. Subject to the assumption that the map v bar right arrow A(v)v, v is an element of R-d, is Lipschitz continuous and strongly monotone, it is proved that the proposed method is well-posed. A priori error estimates are presented of the error in the broken H-1(Omega)-norm, exhibiting precisely the same h-optimal and mildly p-suboptimal convergence rates as obtained for the interior penalty approximation of linear elliptic problems. A priori estimates for linear functionals of the error and the L-2(Omega)-norm of the error are also established and shown to be h-optimal for a particular member of the proposed family of methods. The analysis is completed under fairly weak conditions on the approximation space, allowing for non-affine and curved elements with multilevel hanging nodes. The theoretical results are verified by numerical experiments.

• 出版日期2015