Abstract. 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 Galerkin methods is developed using only the H k weak formulation of a boundary value problem of order 2k. This is accomplished by replacing the Galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in H k .

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Abstract. 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 well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.

In this paper, we have analyzed a one parameter family of hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems −∇ · a(u, ∇u) + f (u, ∇u) = 0 with Dirichlet boundary conditions. These methods depend on the values of the parameter θ ∈ [−1, 1], where θ = +1 corresponds to the nonsymmetric and θ = −1 corresponds to the symmetric interior penalty methods when a(u, ∇u) = ∇u and f (u, ∇u) = − f , that is, for the Poisson problem. The error estimate in the broken H 1 norm, which is optimal in h (mesh size) and suboptimal in p (degree of approximation) is derived using piecewise polynomials of degree p ≥ 2, when the solution u ∈ H 5/2 (Ω). In the case of linear elliptic problems also, this estimate is optimal in h and suboptimal in p. Further, optimal error estimate in the L 2 norm when θ = −1 is derived. Numerical experiments are presented to illustrate the theoretical results. Mathematics Subject Classification (2000) 65N12 · 65N30 · 65N15Supported by DST-DAAD (PPP-05) project.

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