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A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces

David Doyen, Alexandre Ern, Serge Piperno (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented Lagrangian....

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Gunar Matthies, Piotr Skrzypacz, Lutz Tobiska (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation and projection spaces...

A viscosity solution method for Shape-From-Shading without image boundary data

Emmanuel Prados, Fabio Camilli, Olivier Faugeras (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we propose a solution of the Lambertian shape-from-shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solution. The power of our approach is twofolds: (1) it defines a notion of weak solutions (in the viscosity sense) which does not necessarily require boundary data. Moreover, it allows to characterize the viscosity solutions by their “minimums”; and (2) it unifies the works of [Rouy and Tourin, SIAM J. Numer. Anal.29 (1992) 867–884],...

Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem

Xinlong Feng, Zhifeng Weng, Hehu Xie (2014)

Applications of Mathematics

This paper provides an accelerated two-grid stabilized mixed finite element scheme for the Stokes eigenvalue problem based on the pressure projection. With the scheme, the solution of the Stokes eigenvalue problem on a fine grid is reduced to the solution of the Stokes eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. By solving a slightly different linear problem on the fine grid, the new algorithm significantly improves the theoretical error...

Adaptive finite element methods for elliptic problems: Abstract framework and applications

Serge Nicaise, Sarah Cochez-Dhondt (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension not necessarily included into V. We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems...

Adaptivity and variational stabilization for convection-diffusion equations

Albert Cohen, Wolfgang Dahmen, Gerrit Welper (2012)

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

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...

Adaptivity and variational stabilization for convection-diffusion equations∗

Albert Cohen, Wolfgang Dahmen, Gerrit Welper (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...

Algebraic domain decomposition solver for linear elasticity

Aleš Janka (1999)

Applications of Mathematics

We generalize the overlapping Schwarz domain decomposition method to problems of linear elasticity. The convergence rate independent of the mesh size, coarse-space size, Korn’s constant and essential boundary conditions is proved here. Abstract convergence bounds developed here can be used for an analysis of the method applied to singular perturbations of other elliptic problems.

An alternating-direction iteration method for Helmholtz problems

Jim Douglas, Jeffrey L. Hensley, Jean Elizabeth Roberts (1993)

Applications of Mathematics

An alternating-direction iterative procedure is described for a class of Helmholz-like problems. An algorithm for the selection of the iteration parameters is derived; the parameters are complex with some having positive real part and some negative, reflecting the noncoercivity and nonsymmetry of the finite element or finite difference matrix. Examples are presented, with an applications to wave propagation.

An analysis of the boundary layer in the 1D surface Cauchy–Born model

Kavinda Jayawardana, Christelle Mordacq, Christoph Ortner, Harold S. Park (2013)

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

The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error...

An analysis of the boundary layer in the 1D surface Cauchy–Born model∗

Kavinda Jayawardana, Christelle Mordacq, Christoph Ortner, Harold S. Park (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error...

An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows

Andrea Manzoni (2014)

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

We present the current Reduced Basis framework for the efficient numerical approximation of parametrized steady Navier–Stokes equations. We have extended the existing setting developed in the last decade (see e.g. [S. Deparis, SIAM J. Numer. Anal. 46 (2008) 2039–2067; A. Quarteroni and G. Rozza, Numer. Methods Partial Differ. Equ. 23 (2007) 923–948; K. Veroy and A.T. Patera, Int. J. Numer. Methods Fluids 47 (2005) 773–788]) to more general affine and nonaffine parametrizations (such as volume-based...

An hp-Discontinuous Galerkin Method for the Optimal Control Problem of Laser Surface Hardening of Steel

Gupta Nupur, Nataraj Neela (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we discuss an hp-discontinuous Galerkin finite element method (hp-DGFEM) for the laser surface hardening of steel, which is a constrained optimal control problem governed by a system of differential equations, consisting of an ordinary differential equation for austenite formation and a semi-linear parabolic differential equation for temperature evolution. The space discretization of the state variable is done using an hp-DGFEM, time and control discretizations are based on a discontinuous Galerkin...

An hp-Discontinuous Galerkin Method for the Optimal Control Problem of Laser Surface Hardening of Steel

Gupta Nupur, Nataraj Neela (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we discuss an hp-discontinuous Galerkin finite element method (hp-DGFEM) for the laser surface hardening of steel, which is a constrained optimal control problem governed by a system of differential equations, consisting of an ordinary differential equation for austenite formation and a semi-linear parabolic differential equation for temperature evolution. The space discretization of the state variable is done using an hp-DGFEM, time and control discretizations are based on a discontinuous Galerkin...

An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model

Laura Gastaldo, Raphaèle Herbin, Jean-Claude Latché (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of...

An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations

Thierry Gallouët, Laura Gastaldo, Raphaele Herbin, Jean-Claude Latché (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution. The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L2-stability of the discrete advection operator provided it...

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