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Impact of the variations of the mixing length in a first order turbulent closure system

Françoise Brossier, Roger Lewandowski (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is devoted to the study of a turbulent circulation model. Equations are derived from the “Navier-Stokes turbulent kinetic energy” system. Some simplifications are performed but attention is focused on non linearities linked to turbulent eddy viscosity  ν t . The mixing length ℓ acts as a parameter which controls the turbulent part in ν t . The main theoretical results that we have obtained concern the uniqueness of the solution for bounded eddy viscosities and small values of ℓ and its asymptotic...

Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficients⋆⋆⋆

J. Beck, F. Nobile, L. Tamellini, R. Tempone (2011)

ESAIM: Proceedings

In this work we first focus on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE. We rely on sharp estimates for the decay of the coefficients of the spectral expansion of u on orthogonal polynomials to build a sequence of polynomial subspaces that features better convergence properties compared to standard polynomial subspaces such as Total Degree or Tensor Product. We consider then the Stochastic Collocation method, and use the previous estimates to introduce...

Implicit a posteriori error estimation using patch recovery techniques

Tamás Horváth, Ferenc Izsák (2012)

Open Mathematics

We develop implicit a posteriori error estimators for elliptic boundary value problems. Local problems are formulated for the error and the corresponding Neumann type boundary conditions are approximated using a new family of gradient averaging procedures. Convergence properties of the implicit error estimator are discussed independently of residual type error estimators, and this gives a freedom in the choice of boundary conditions. General assumptions are elaborated for the gradient averaging...

Improved convergence bounds for smoothed aggregation method: linear dependence of the convergence rate on the number of levels

Jan Brousek, Pavla Fraňková, Petr Vaněk (2016)

Czechoslovak Mathematical Journal

The smoothed aggregation method has became a widely used tool for solving the linear systems arising by the discretization of elliptic partial differential equations and their singular perturbations. The smoothed aggregation method is an algebraic multigrid technique where the prolongators are constructed in two steps. First, the tentative prolongator is constructed by the aggregation (or, the generalized aggregation) method. Then, the range of the tentative prolongator is smoothed by a sparse linear...

Improved flux reconstructions in one dimension

Vlasák, Miloslav, Lamač, Jan (2023)

Programs and Algorithms of Numerical Mathematics

We present an improvement to the direct flux reconstruction technique for equilibrated flux a posteriori error estimates for one-dimensional problems. The verification of the suggested reconstruction is provided by numerical experiments.

Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell's problem

Yanlai Chen, Jan S. Hesthaven, Yvon Maday, Jerónimo Rodríguez (2009)

ESAIM: Mathematical Modelling and Numerical Analysis


In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-sup stability constants is essential. In [Huynh et al., C. R. Acad. Sci. Paris Ser. I Math.345 (2007) 473–478], the authors presented an efficient method, compatible with an off-line/on-line strategy, where the on-line computation is reduced to minimizing a linear functional under a few linear constraints. These constraints...

Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra

Gunar Matthies (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We present families of scalar nonconforming finite elements of arbitrary order r ≥ 1 with optimal approximation properties on quadrilaterals and hexahedra. Their vector-valued versions together with a discontinuous pressure approximation of order r - 1 form inf-sup stable finite element pairs of order r for the Stokes problem. The well-known elements by Rannacher and Turek are recovered in the case r=1. A numerical comparison between conforming and nonconforming discretisations will be given. Since higher order...

Inner products in covolume and mimetic methods

Kathryn A. Trapp (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This...

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