Thirupathi Gudi

A new error analysis for discontinuous finite element methods for linear elliptic problems

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 .

{C}^0

In this paper, we develop and analyze C(0) penalty methods for the fully nonlinear Monge-Ampere 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.

penalty methods for the fully nonlinear Monge-Ampère equation

A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617–637, 2005) and Carstensen and Hu (J Numer Math 107(3):473–502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied to the Laplace, Stokes, and Lamé equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature. The estimate recovers known results as well as provides new bounds to a number of schemes.

A unifying theory of a posteriori error control for discontinuous Galerkin FEM

In this paper, both symmetric and nonsymmetric interior penalty discontinuous

Discontinuous Galerkin Methods for Quasi-Linear Elliptic Problems of Nonmonotone Type

hp

A Quadratic

-Galerkin methods are applied to a class of quasi-linear elliptic problems which are of nonmonotone type. Using Brouwer’s 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 also proved. A priori error estimates in the broken

C^0

H^1

Interior Penalty Method for Linear Fourth Order Boundary Value Problems with Boundary Conditions of the Cahn--Hilliard Type

-norm, which are optimal in

Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes

h

Finite Element Method for a Nonlocal Problem of Kirchhoff Type

and suboptimal in