69 research outputs found
Mixed finite element methods for stationary incompressible magneto-hydrodynamics
Summary.: A new mixed variational formulation of the equations of stationary incompressible magneto-hydrodynamics is introduced and analyzed. The formulation is based on curl-conforming Sobolev spaces for the magnetic variables and is shown to be well-posed in (possibly non-convex) Lipschitz polyhedra. A finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf-sup stable velocity-pressure space pairs and the magnetic ones by a mixed approach using Nédélec's elements of the first kind. An error analysis is carried out that shows that the proposed finite element approximation leads to quasi-optimal error bounds in the mesh-siz
Interior penalty discontinuous Galerkin method for Maxwell's equations: optimal L2-norm error estimates
We consider the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form. In Grote et al. (2007, J. Comput. Appl. Math., 204, 375-386), optimal a priori estimates in the DG energy norm were derived, either for smooth solutions on arbitrary meshes or for low-regularity (singular) solutions on conforming, affine meshes. Here, we show that the DG methods are also optimally convergent in the L2-norm, on tetrahedral meshes and for smooth material coefficients. The theoretical convergence rates are validated by a series of numerical experiments in two-space dimensions, which also illustrate the usefulness of the interior penalty DG method for time-dependent computational electromagnetic
A posteriori error estimation for hp -version time-stepping methods for parabolic partial differential equations
The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiment
Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation
In Grote et al. (SIAM J.Numer.Anal., 44:2408-2431, 2006) a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L 2-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation ("leap-frogâ scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1+Ît 2), where p denotes the polynomial degree, h the mesh size, and Ît the time ste
Exponential convergence of mixed hp-DGFEM for Stokes flow in polygons
Summary.: We analyze mixed hp-discontinuous Galerkin finite element methods (DGFEM) for Stokes flow in polygonal domains. In conjunction with geometrically refined quadrilateral meshes and linearly increasing approximation orders, we prove that the hp-DGFEM leads to exponential rates of convergence for piecewise analytic solutions exhibiting singularities near corner
Mixed hpâDGFEM for incompressible flows II: Geometric edge meshes
We consider the Stokes problem of incompressible fluid flow in threeâdimensional polyhedral domains discretized on hexahedral meshes with hpâdiscontinuous Galerkin finite elements of type Qk for the velocity and Qkâ1 for the pressure. We prove that these elements are infâsup stable on geometric edge meshes that are refined anisotropically and nonâquasiuniformly towards edges and corners. The discrete infâsup constant is shown to be independent of the aspect ratio of the anisotropic elements and is of O(kâ3/2) in the polynomial degree k, as in the case of conforming QkâQkâ2 approximations on the same meshe
A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations
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 present a set of numerical tests that confirm these properties. The results of this note naturally expand the work in a previous publication on Navier-Stokes equation
Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates
AbstractWe develop the symmetric interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell equations in second-order form. We derive optimal a priori error estimates in the energy norm for smooth solutions. We also consider the case of low-regularity solutions that have singularities in space
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