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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...
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 norm for the sequence of discrete operators....
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...
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...
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...
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 ), soit en augmentant l’ordre de consistance (analogue à la stratégie de raffinement ). Néanmoins, le coût du calcul des fonctions d’interpolation est très...
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...
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.
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...
We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain 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 . 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...
We consider a nonlinear second order elliptic boundary
value problem (BVP)
in a bounded domain 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...
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....
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...
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 norms obtained by Nečas and on the general
framework of Γ-convergence theory.
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