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The standard a priori error analysis of discontinuous Galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the Galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm. In this paper, a new error analysis of discontinuous… (More)

In this paper, we develop and analyze C 0 penalty methods for the fully nonlinear Monge-Ampère equation det(D 2 u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the… (More)

In this paper, we first split the biharmonic equation 2 u = f with nonhomoge-neous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v = u and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation v h of v can easily be eliminated to reduce the… (More)

We study a class of symmetric discontinuous Galerkin methods on graded meshes. Optimal order error estimates are derived in both the energy norm and the L 2 norm, and we establish the uniform convergence of V-cycle, F-cycle and W-cycle multigrid algorithms for the resulting discrete problems. Numerical results that confirm the theoretical results are also… (More)

In this paper, an hp-local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On hp-quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is… (More)