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𝐴 - 𝑃𝑂𝑆𝑇𝐸𝑅𝐼𝑂𝑅𝐼 error estimates for linear exterior problems 𝑉𝐼𝐴 mixed-FEM and DtN mappings

Mauricio A. Barrientos, Gabriel N. Gatica, Matthias Maischak (2002)

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

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...

2-dimensional primal domain decomposition theory in detail

Dalibor Lukáš, Jiří Bouchala, Petr Vodstrčil, Lukáš Malý (2015)

Applications of Mathematics

We give details of the theory of primal domain decomposition (DD) methods for a 2-dimensional second order elliptic equation with homogeneous Dirichlet boundary conditions and jumping coefficients. The problem is discretized by the finite element method. The computational domain is decomposed into triangular subdomains that align with the coefficients jumps. We prove that the condition number of the vertex-based DD preconditioner is O ( ( 1 + log ( H / h ) ) 2 ) , independently of the coefficient jumps, where H and h denote...

3D domain decomposition method coupling conforming and nonconforming finite elements

Abdellatif Agouzal, Laurence Lamoulie, Jean-Marie Thomas (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper deals with the solution of problems involving partial differential equations in  ℝ 3 . For three dimensional case, methods are useful if they require neither domain boundary regularity nor regularity for the exact solution of the problem. A new domain decomposition method is therefore presented which uses low degree finite elements. The numerical approximation of the solution is easy, and optimal error bounds are obtained according to suitable norms.

A balanced finite-element method for an axisymmetrically loaded thin shell

Norbert Heuer, Torsten Linss (2024)

Applications of Mathematics

We analyse a finite-element discretisation of a differential equation describing an axisymmetrically loaded thin shell. The problem is singularly perturbed when the thickness of the shell becomes small. We prove robust convergence of the method in a balanced norm that captures the layers present in the solution. Numerical results confirm our findings.

A Bermúdez–Moreno algorithm adapted to solve a viscoplastic problem in alloy solidification processes

P. Barral, P. Quintela, M. T. Sánchez (2014)

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

The aim of this work is to present a computationally efficient algorithm to simulate the deformations suffered by a viscoplastic body in a solidification process. This type of problems involves a nonlinearity due to the considered thermo-elastic-viscoplastic law. In our previous papers, this difficulty has been solved by means of a duality method, known as Bermúdez–Moreno algorithm, involving a multiplier which was computed with a fixed point algorithm or a Newton method. In this paper, we will...

A boundary multivalued integral “equation” approach to the semipermeability problem

Jaroslav Haslinger, Charalambos C. Baniotopoulos, Panagiotis D. Panagiotopoulos (1993)

Applications of Mathematics

The present paper concerns the problem of the flow through a semipermeable membrane of infinite thickness. The semipermeability boundary conditions are first considered to be monotone; these relations are therefore derived by convex superpotentials being in general nondifferentiable and nonfinite, and lead via a suitable application of the saddlepoint technique to the formulation of a multivalued boundary integral equation. The latter is equivalent to a boundary minimization problem with a small...

A C1-P2 finite element without nodal basis

Shangyou Zhang (2008)

ESAIM: Mathematical Modelling and Numerical Analysis


A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic...

A class of nonparametric DSSY nonconforming quadrilateral elements

Youngmok Jeon, Hyun NAM, Dongwoo Sheen, Kwangshin Shim (2013)

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

A new class of nonparametric nonconforming quadrilateral finite elements is introduced which has the midpoint continuity and the mean value continuity at the interfaces of elements simultaneously as the rectangular DSSY element [J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, ESAIM: M2AN 33 (1999) 747–770.] The parametric DSSY element for general quadrilaterals requires five degrees of freedom to have an optimal order of convergence [Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Calcolo...

A comparison of dual Lagrange multiplier spaces for Mortar finite element discretizations

Barbara I. Wohlmuth (2002)

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

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We focus on mortar finite element methods on non-matching triangulations. In particular, we discuss and analyze dual Lagrange multiplier spaces for lowest order finite elements. These non standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a consequence, standard efficient iterative...

A Comparison of Dual Lagrange Multiplier Spaces for Mortar Finite Element Discretizations

Barbara I. Wohlmuth (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We focus on mortar finite element methods on non-matching triangulations. In particular, we discuss and analyze dual Lagrange multiplier spaces for lowest order finite elements. These non standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a consequence, standard efficient iterative...

A comparison of some a posteriori error estimates for fourth order problems

Segeth, Karel (2010)

Programs and Algorithms of Numerical Mathematics

A lot of papers and books analyze analytical a posteriori error estimates from the point of view of robustness, guaranteed upper bounds, global efficiency, etc. At the same time, adaptive finite element methods have acquired the principal position among algorithms for solving differential problems in many physical and technical applications. In this survey contribution, we present and compare, from the viewpoint of adaptive computation, several recently published error estimation procedures for...

A comparison of some efficient numerical methods for a nonlinear elliptic problem

Balázs Kovács (2012)

Open Mathematics

The aim of this paper is to compare and realize three efficient iterative methods, which have mesh independent convergence, and to propose some improvements for them. We look for the numerical solution of a nonlinear model problem using FEM discretization with gradient and Newton type methods. Three numerical methods have been carried out, namely, the gradient, Newton and quasi-Newton methods. We have solved the model problem with these methods, we have investigated the differences between them...

A comparison of two FEM-based methods for the solution of the nonlinear output regulation problem

Branislav Rehák, Sergej Čelikovský, Javier Ruiz, Jorge Orozco-Mora (2009)

Kybernetika

The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper...

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