69 research outputs found
Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements
The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical
Methods for Singularly Perturbed Differential Equations" appeared many years
ago and was for many years a reliable guide into the world of numerical methods
for singularly perturbed problems. Since then many new results came into the
game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827
Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?
The contents of this paper is twofold. First, important recent results concerning finite element
methods for convection-dominated problems and incompressible flow problems are described that
illustrate the activities in these topics. Second, a number of, in our opinion, important problems in
these fields are discussed
Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
International audienceWe derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart-Thomas space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincaré, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates
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Computational Engineering
The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods
Anisotropic a posteriori error estimation for the mixed discontinuous Galerkin approximation of the Stokes problem
International audienceThe paper presents a posteriori error estimates for the mixed discontinuous Galerkin approximation of the stationary Stokes problem. We consider anisotropic finite element discretizations, i.e. elements with very large aspect ratio. Our analysis covers two- and three-dimensional domains. Lower and upper error bounds are proved with minimal assumptions on the meshes. The lower error bound is uniform with respect to the mesh anisotropy. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimator
Numerical solution of steady-state groundwater flow and solute transport problems: Discontinuous Galerkin based methods compared to the Streamline Diffusion approach
In this study, we consider the simulation of subsurface flow and solute
transport processes in the stationary limit. In the convection-dominant case,
the numerical solution of the transport problem may exhibit non-physical
diffusion and under- and overshoots. For an interior penalty discontinuous
Galerkin (DG) discretization, we present a -adaptive refinement strategy
and, alternatively, a new efficient approach for reducing numerical under- and
overshoots using a diffusive -projection. Furthermore, we illustrate an
efficient way of solving the linear system arising from the DG discretization.
In -D and -D examples, we compare the DG-based methods to the streamline
diffusion approach with respect to computing time and their ability to resolve
steep fronts
Stability of an upwind Petrov Galerkin discretization of convection diffusion equations
We study a numerical method for convection diffusion equations, in the regime
of small viscosity. It can be described as an exponentially fitted conforming
Petrov-Galerkin method. We identify norms for which we have both continuity and
an inf-sup condition, which are uniform in mesh-width and viscosity, up to a
logarithm, as long as the viscosity is smaller than the mesh-width or the
crosswind diffusion is smaller than the streamline diffusion. The analysis
allows for the formation of a boundary layer.Comment: v1: 18 pages. 2 figures. v2: 22 pages. Numerous details added and
completely rewritten final proof. 8 pages appendix with old proo
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