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A penalty method for the time-dependent Stokes problem with the slip boundary condition and its finite element approximation

Guanyu Zhou, Takahito Kashiwabara, Issei Oikawa (2017)

Applications of Mathematics

We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is introduced, which also facilitates the numerical implementation. For the continuous problem, the convergence of the penalty method is investigated. Then we study the fully discretized finite element approximations for the penalty method with the P1/P1-stabilization or P1b/P1 element. For the discretization...

A Petrov-Galerkin approximation of convection-diffusion and reaction-diffusion problems

Josef Dalík (1991)

Applications of Mathematics

A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation - ϵ u n + p u ' + q u = f are presented and analyzed theoretically. The positive number ϵ is supposed to be much less than the discretization step and the values of p , q . An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.

A piecewise P2-nonconforming quadrilateral finite element

Imbunm Kim, Zhongxuan Luo, Zhaoliang Meng, Hyun NAM, Chunjae Park, Dongwoo Sheen (2013)

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

We introduce a piecewise P2-nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P2-polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the...

A posteriori error analysis for parabolic variational inequalities

Kyoung-Sook Moon, Ricardo H. Nochetto, Tobias von Petersdorff, Chen-song Zhang (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain Ω ⊂ ℝ d with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L2(0,T;H1(Ω)). The error estimator is localized in the sense that the size of the elliptic residual...

A posteriori error analysis of the fully discretized time-dependent Stokes equations

Christine Bernardi, Rüdiger Verfürth (2004)

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

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

A posteriori error analysis of the fully discretized time-dependent Stokes equations

Christine Bernardi, Rüdiger Verfürth (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

A posteriori error estimates for a nonconforming finite element discretization of the heat equation

Serge Nicaise, Nadir Soualem (2005)

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

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in ℝ d , d = 2 or 3, using backward Euler’s scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results...

A posteriori error estimates for a nonconforming finite element discretization of the heat equation

Serge Nicaise, Nadir Soualem (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in ℝ d , d=2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main...

A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces

Juan Pablo Agnelli, Eduardo M. Garau, Pedro Morin (2014)

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

In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation...

A Posteriori Error Estimates for Finite Volume Approximations

S. Cochez-Dhondt, S. Nicaise, S. Repin (2009)

Mathematical Modelling of Natural Phenomena

We present new a posteriori error estimates for the finite volume approximations of elliptic problems. They are obtained by applying functional a posteriori error estimates to natural extensions of the approximate solution and its flux computed by the finite volume method. The estimates give guaranteed upper bounds for the errors in terms of the primal (energy) norm, dual norm (for fluxes), and also in terms of the combined primal-dual norms. It is shown that the estimates provide sharp upper and...

A posteriori error estimates for linear exterior problems via mixed-FEM and DtN mappings

Mauricio A. Barrientos, Gabriel N. Gatica, Matthias Maischak (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the...

A posteriori error estimates for the 3 D stabilized Mortar finite element method applied to the Laplace equation

Zakaria Belhachmi (2003)

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

We consider a non-conforming stabilized domain decomposition technique for the discretization of the three-dimensional Laplace equation. The aim is to extend the numerical analysis of residual error indicators to this model problem. Two formulations of the problem are considered and the error estimators are studied for both. In the first one, the error estimator provides upper and lower bounds for the energy norm of the mortar finite element solution whereas in the second case, it also estimates...

A posteriori Error Estimates For the 3D Stabilized Mortar Finite Element Method applied to the Laplace Equation

Zakaria Belhachmi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a non-conforming stabilized domain decomposition technique for the discretization of the three-dimensional Laplace equation. The aim is to extend the numerical analysis of residual error indicators to this model problem. Two formulations of the problem are considered and the error estimators are studied for both. In the first one, the error estimator provides upper and lower bounds for the energy norm of the mortar finite element solution whereas in the second case, it also estimates...

A Posteriori Error Estimates on Stars for Convection Diffusion Problem

B. Achchab, A. Agouzal, K. Bouihat (2010)

Mathematical Modelling of Natural Phenomena

In this paper, a new a posteriori error estimator for nonconforming convection diffusion approximation problem, which relies on the small discrete problems solution in stars, has been established. It is equivalent to the energy error up to data oscillation without any saturation assumption nor comparison with residual estimator

A posteriori error estimates with post-processing for nonconforming finite elements

Friedhelm Schieweck (2002)

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

For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution. Thus, for the error analysis, the existing theory from the conforming case can be used together with some simple additional arguments. As an essential point, the property is exploited...

A posteriori Error Estimates with Post-Processing for Nonconforming Finite Elements

Friedhelm Schieweck (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution. Thus, for the error analysis, the existing theory from the conforming case can be used together with some simple additional arguments. As an essential point, the property is...

Currently displaying 161 – 180 of 374