Papers
Minimal stabilization of discontinuous Galerkin finite element methods for hyperbolic problems
Co-authored with E. Burman; Journal of Scientific Computing, vol. 33, num. 2, 2007, p. 183-208
We consider a discontinuous Galerkin finite element method for
the advection–reaction equation in two space–dimensions. For polynomial approximation spaces of degree greater than or equal to two on triangles we propose a method where stability is obtained by a penalization of only the upper portion of the polynomial spectrum of the jump of the solution over element edges. We prove stability in the standard h-weighted graphnorm and obtain optimal order error estimates with respect to mesh-size.
LikeThe symmetric discontinuous Galerkin method does not need stabilization in 1D for polynomial orders p ≥ 2
Co-authored with E. Burman, A. Ern and I. Mozolevski; C. R. Math. Acad. Sci. Paris, vol. 345, num. 10, 2007, p. 599-602
In this Note we prove that in one space dimension, the symmetric discontinuous Galerkin method for second order elliptic problems is stable for polynomial orders without using any stabilization parameter. The method yields optimal convergence rates in both the broken energy norm and the L2-norm and can be written in conservative form with fluxes independent of any stabilization parameter.
Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment
Co-authored with E. Burman; C. R. Math. Acad. Sci. Paris, vol. 346, num. 1-2, 2008, p. 103-106
n this Note we prove that in two and three space dimensions, the symmetric and non-symmetric discontinuous Galerkin method for second order elliptic problems is stable when using piecewise linear elements enriched with quadratic bubbles without any penalization of the interelement jumps. The method yields optimal convergence rates in both the broken energy norm and, in the symmetric case, the L2-norm. Moreover the method can be written in conservative form with fluxes independent of any stabilization parameter.
Stabilization strategies for high order methods for transport dominated problems
Co-authored with E. Burman and A. Quarteroni; Bollettino U.M.I., Series IX, vol. 1, num. 1, 2008, p. 57-77
Standard high order Galerkin methods, such as pure spectral or high order finite element methods, have insufficient stability properties when applied to transport dominated problems. In this paper we review some stabilization strategies for pure spectral methods and spectral multidomain approaches.
Local discontinuous Galerkin method for diffusion equations with reduced stabilization
Co-authored with E. Burman; Commun. Comput. Phys., vol. 5, 2009, p. 498-514
We extend the results on minimal stabilization of Burman and Stamm[J. Sci. Comp., 33 (2007), pp. 183-208] to the case of the local discontinuous Galerkin methods on mixed form. The penalization term on the faces is relaxed to act only on a part of the polynomial spectrum. Stability in the form of a discrete inf-sup condition is proved and optimal convergence follows. Some numerical examples using high order approximation spaces illustrate the theory.
Interior Penalty continuous and discontinuous finite element approximations of hyperbolic equations
Co-authored with E. Burman and A. Quarteroni; accepted in Journal of Scientific Computing
In this paper we present in a unified setting the continuous and discontinuous Galerkin methods for the numerical approximation of the scalar hyperbolic equation. Both methods are stabilized by the interior penalty method, more precisely by the jump of the gradient across element faces in the continuous case whereas in the discontinuous case the stabilization of the jump of the solution and optionally of its gradient is required to achieve optimal convergence. We prove that the solution in the case of the continuous Galerkin approach can be considered as a limit of the discontinuous one when the stabilization parameter associated with the penalization of the solution jump tends to infinity. As a consequence, the limit of the numerical flux of the discontinuous method yields a numerical flux for the continuous method as well. Numerical results will highlight the theoretical results that are proven in this paper.
hp-Optimal Discontinuous Galerkin Methods for Linear Elliptic Problems
Co-authored with Th. P. Wihler; accepted in Mathematics of Computation
The aim of this paper is to present and analyze a class of hp-version discontinuous Galerkin (DG) discretizations for the numerical approximation of linear elliptic problems. This class includes a number of well-known DG formulations. We will show that the methods are stable provided that the stability parameters are suitably chosen. Furthermore, on (possibly irregular) quadrilateral meshes, we shall prove that the schemes converge all optimally in the energy norm with respect to both the local element sizes and polynomial degrees provided that homogeneous boundary conditions are considered.
Bubble stabilized discontinuous {G}alerkin method for {S}tokes' problem
Co-authered with E. Burman; accepted in Mathematical Models and Methods in Applied Sciences
We propose a low order discontinuous Galerkin method for incompressible flows. Stability of the discretization of the Laplace operator is obtained by enriching the space element wise with a non-conforming quadratic bubble.
Several possible pressure spaces that lead to uniformly stable velocity pressure pairs are proposed. We prove optimal convergence estimates and local conservation of both mass and linear momentum independent of numerical parameters.
Low Order Discontinuous Galerkin Methods for Second Order Elliptic Problems
Co-authored with E. Burman; SIAM J. Numer. Anal., Vol. 47, No. 1, pp. 508-533 (2008)
We consider DG-methods for second order scalar elliptic problems using piecewise affine approximation in two or three space dimensions. We prove that both the symmetric and the nonsymmetric versions of the DG-method have regular system matrices without penalization of the interelement solution jumps provided boundary conditions are imposed in a certain weak manner. Optimal convergence is proved for sufficiently regular meshes and data. We then propose a DG-method using piecewise affine functions enriched with quadratic bubbles. Using this space we prove optimal convergence in the energy norm for both a symmetric and non-symmetric DG-method without stabilization. All of these proposed methods share the feature that they conserve mass locally independent of the penalty parameter.
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