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- Wolfgang Bangerth, Ralf Hartmann, Guido Kanschat
- ACM Trans. Math. Softw.
- 2007

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… (More)

- Jay Gopalakrishnan, Guido Kanschat
- Numerische Mathematik
- 2003

A variable V-cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented. An analysis under a mild regularity assumption shows that the preconditioner is uniform. The interior penalty method is then combined with a discontinuous Galerkin scheme to arrive at a discretization scheme for an advection-diffusion… (More)

- Bernardo Cockburn, Guido Kanschat, Dominik Schötzau, Christoph Schwab
- SIAM J. Numerical Analysis
- 2002

- Bernardo Cockburn, Guido Kanschat, Ilaria Perugia, Dominik Schötzau
- SIAM J. Numerical Analysis
- 2001

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)

- Bernardo Cockburn, Guido Kanschat, Dominik Schötzau
- Math. Comput.
- 2005

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 conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely… (More)

- Bernardo Cockburn, Guido Kanschat, Dominik Schötzau
- J. Sci. Comput.
- 2009

We introduce and analyze a discontinuous Galerkin method for the incompressible Navier-Stokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability of this equal-order approach is ensured by a pressure stabilization term. A simple element-by-element postprocessing… (More)

- Bernardo Cockburn, Guido Kanschat, Dominik Schötzau
- J. Sci. Comput.
- 2007

We present a class of discontinuous Galerkin methods for the incompressible 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 convergent. We… (More)

- Jean-Luc Guermond, Guido Kanschat
- SIAM J. Numerical Analysis
- 2010

We revisit some results from M. L. Adams [Nucl. Sci. Engrg., 137 (2001), pp. 298– 333]. Using functional analytic tools we prove that a necessary and sufficient condition for the standard upwind discontinuous Galerkin approximation to converge to the correct limit solution in the diffusive regime is that the approximation space contains a linear space of… (More)

- Guido Kanschat
- 1998

We derive a posteriori error estimates for nonconforming discretizations of Poisson's and Stokes' equations. The estimates are residual based and make use of weight factors obtained by a duality argument. Crouzeix-Raviart elements on triangles and rotated bilinear elements are considered. The quadrilateral case involves the introduction of additional local… (More)

- Guido Kanschat
- 1998

A stabilized weak formulation of the radiative transfer equation is presented, which is stable independent of the scattering coeecient. This enables the use of standard nite element discretizations without further algebraic constraints. Furthermore, a weighted residual-based a posteriori error estimate is derived for the discrete solution. An example… (More)