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Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd

M. Aurada, M. Feischl, J. Kemetmüller, M. Page, D. Praetorius (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the solution of second order elliptic PDEs in Rd with inhomogeneous Dirichlet data by means of an h–adaptive FEM with fixed polynomial order p ∈ N. As model example serves the Poisson equation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an H1 / 2–stable projection, for instance, the L2–projection for p = 1 or the Scott–Zhang projection for general p ≥ 1. For error estimation, we use a residual error estimator which...

Edge finite elements for the approximation of Maxwell resolvent operator

Daniele Boffi, Lucia Gastaldi (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi [14]. We shall review some of the known properties for edge element approximations and prove some new result. In particular we shall prove a uniform convergence in the L 2 norm for the sequence of discrete operators....

Edge finite elements for the approximation of Maxwell resolvent operator

Daniele Boffi, Lucia Gastaldi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi [14]. We shall review some of the known properties for edge element approximations and prove some new result. In particular we shall prove a uniform convergence in the L2 norm for the sequence of discrete...

Efficient numerical solution of mixed finite element discretizations by adaptive multilevel methods

Ronald H.W. Hoppe, Barbara Wohlmuth (1995)

Applications of Mathematics

We consider mixed finite element discretizations of second order elliptic boundary value problems. Emphasis is on the efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations. In particular, we present two multilevel solvers, the first one relying on ideas from domain decomposition and the second one resulting from mixed hybridization. Local refinement of the underlying triangulations is done by efficient and reliable a...

Electrowetting of a 3D drop: numerical modelling with electrostatic vector fields

Patrick Ciarlet Jr., Claire Scheid (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable...

Enrichissement des interpolations d’éléments finis en utilisant des méthodes sans maillage

Antonio Huerta, Sonia Fernández-Méndez, Pedro Díez (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d’une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement h ), soit en augmentant l’ordre de consistance (analogue à la stratégie de raffinement p ). Néanmoins, le coût du calcul des fonctions d’interpolation est très...

Enrichissement des interpolations d'éléments finis en utilisant des méthodes sans maillage

Antonio Huerta, Sonia Fernández-Méndez, Pedro Díez (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie. Un des atouts de ces méthodes est que la discrétisation peut être enrichie d'une façon très simple, soit en augmentant le nombre de particules (analogue à la stratégie de raffinement h), soit en augmentant l'ordre de consistance (analogue à la stratégie de raffinement p). Néanmoins, le coût du calcul des fonctions d'interpolation est...

Error estimates for linear finite elements on Bakhvalov-type meshes

Hans-Görg Roos (2006)

Applications of Mathematics

For convection-diffusion problems with exponential layers, optimal error estimates for linear finite elements on Shishkin-type meshes are known. We present the first optimal convergence result in an energy norm for a Bakhvalov-type mesh.

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Annalisa Buffa, Peter Monk (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equation provides a variational framework suitable for discretization using plane wave solutions of an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous...

Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition

Marian Slodička (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain Ω ⊂ ℝ dim with a nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant, accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part Γ n . The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the linearization...

Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition

Marian Slodička (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain Ω ⊂ ℝ N with a nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant, accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part Γn. The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for...

Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows

Marco Picasso, Jacques Rappaz (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a convex polygon, existence of a solution is proved for small relaxation times. Continuous piecewise linear finite elements together with a Galerkin Least Square (GLS) method are studied for solving this problem. Existence and a priori error estimates are established using a Newton-chord fixed point theorem, a posteriori error estimates are also derived....

Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows

Marco Picasso, Jacques Rappaz (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a convex polygon, existence of a solution is proved for small relaxation times. Continuous piecewise linear finite elements together with a Galerkin Least Square (GLS) method are studied for solving this problem. Existence and a priori error estimates are established using a Newton-chord fixed point theorem, a posteriori error estimates are also derived. An...

External approximation of first order variational problems via W-1,p estimates

Cesare Davini, Roberto Paroni (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Here we present an approximation method for a rather broad class of first order variational problems in spaces of piece-wise constant functions over triangulations of the base domain. The convergence of the method is based on an inequality involving W - 1 , p norms obtained by Nečas and on the general framework of Γ-convergence theory.

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