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We present a class of discontinuous Galerkin methods for the incom-pressible Navier-Stokes equations yielding exactly divergence-free solutions. Exact incompressibility is achieved by using divergence-conforming velocity spaces for the approximation of the velocities. The resulting methods are locally conservative, energy-stable, and optimally conver-gent.… (More)

In this paper, we present a super-convergence result for the Local Discontinuous Galerkin method for a model elliptic problem on Cartesian grids. We identify a special numerical ux for which the L 2-norm of the gradient and the L 2-norm of the potential are of order k + 1=2 and k + 1, respectively, when tensor product polynomials of degree at most k are… (More)

An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced object-oriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be arranged to fit users requirements. Through this approach,… (More)

In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local con-servativity, its high-order accuracy, and the exact satisfaction of the incom-pressibility constraint. Although the method uses completely… (More)

A local a priori and a posteriori analysis is developed for the Galerkin method with discontinuous finite elements for solving stationary diffusion problems. The main results are an optimal-order estimate for the point-wise error and a corresponding a posteriori error bound. The proofs are based on weighted ¡ £ ¢-norm error estimates for discrete Green… (More)

We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shape-regular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in L 2-and negative-order norms. Numerical experiments are presented which verify these theoretical… (More)

The coupled Stokes and Darcy equations are approximated by a strongly conservative finite element method. The discrete spaces are the divergence-conforming velocity space with matching pressure space such as the Raviart-Thomas spaces. This work proves optimal error estimate of the velocity in the L 2 norm in the domain and on the interface under weak… (More)