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A multiscale mortar multipoint flux mixed finite element method

Mary Fanett Wheeler, Guangri Xue, Ivan Yotov (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cell-centered finite differences on irregular grids. The subdomain grids do not have to match across the interfaces. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid...

A multiscale mortar multipoint flux mixed finite element method

Mary Fanett Wheeler, Guangri Xue, Ivan Yotov (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cell-centered finite differences on irregular grids. The subdomain grids do not have to match across the interfaces. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid...

A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations

Yun-Bo Yang, Qiong-Xiang Kong (2017)

Applications of Mathematics

A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations is presented. Applying the orthogonal projection technique, we introduce two local Gauss integrations as a stabilizing term in the error correction method, and derive a new error correction method. In both the coarse solution computation step and the error computation step, a locally stabilizing term based on two local Gauss integrations is introduced. The stability and convergence of the...

A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method

Abdallah Bradji, Jürgen Fuhrmann (2014)

Mathematica Bohemica

Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of 𝒲 1 , ∞ ( ℒ 2 ) is proved. An ℒ ∞ ( ℋ 1 ) -error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations...

A new finite element approach for problems containing small geometric details

Wolfgang Hackbusch, Stefan A. Sauter (1998)

Archivum Mathematicum

In this paper a new finite element approach is presented which allows the discretization of PDEs on domains containing small micro-structures with extremely few degrees of freedom. The applications of these so-called Composite Finite Elements are two-fold. They allow the efficient use of multi-grid methods to problems on complicated domains where, otherwise, it is not possible to obtain very coarse discretizations with standard finite elements. Furthermore, they provide a tool for discrete homogenization...

A new formulation of the Stokes problem in a cylinder, and its spectral discretization

Nehla Abdellatif, Christine Bernardi (2004)

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

We analyze a new formulation of the Stokes equations in three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to the angular variable: the problem for each Fourier coefficient is two-dimensional and has six scalar unknowns, corresponding to the vector potential and the vorticity. A spectral discretization is built on this formulation, which leads to an exactly divergence-free discrete velocity. We prove optimal error estimates.

A new formulation of the Stokes problem in a cylinder, and its spectral discretization

Nehla Abdellatif, Christine Bernardi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze a new formulation of the Stokes equations in three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to the angular variable: the problem for each Fourier coefficient is two-dimensional and has six scalar unknowns, corresponding to the vector potential and the vorticity. A spectral discretization is built on this formulation, which leads to an exactly divergence-free discrete velocity. We prove optimal error estimates.

A new H(div)-conforming p-interpolation operator in two dimensions

Alexei Bespalov, Norbert Heuer (2011)

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

In this paper we construct a new H(div)-conforming projection-based p-interpolation operator that assumes only Hr(K) ∩ 𝐇 ˜ -1/2(div, K)-regularity (r > 0) on the reference element (either triangle or square) K. We show that this operator is stable...

A new H(div)-conforming p-interpolation operator in two dimensions

Alexei Bespalov, Norbert Heuer (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we construct a new H(div)-conforming projection-based p-interpolation operator that assumes only Hr(K) ∩ 𝐇 ˜ -1/2(div, K)-regularity (r > 0) on the reference element (either triangle or square) K. We show that this operator is stable with...

A note on polynomial approximation in Sobolev spaces

Rüdiger Verfürth (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

For domains which are star-shaped w.r.t. at least one point, we give new bounds on the constants in Jackson-inequalities in Sobolev spaces. For convex domains, these bounds do not depend on the eccentricity. For non-convex domains with a re-entrant corner, the bounds are uniform w.r.t. the exterior angle. The main tool is a new projection operator onto the space of polynomials.

A numerical minimization scheme for the complex Helmholtz equation

Russell B. Richins, David C. Dobson (2012)

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

We use the work of Milton, Seppecher, and Bouchitté on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate...

A numerical minimization scheme for the complex Helmholtz equation

Russell B. Richins, David C. Dobson (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We use the work of Milton, Seppecher, and Bouchitté on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate...

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.

Currently displaying 41 – 60 of 156