The Gauss center research in multiscale scientific computation.
The hierarchical basis multigrid method for convection-diffusion equations.
The mortar element method for three dimensional finite elements
The Mortar finite element method for Bingham fluids
This paper deals with the flow problem of a viscous plastic fluid in a cylindrical pipe. In order to approximate this problem governed by a variational inequality, we apply the nonconforming mortar finite element method. By using appropriate techniques, we are able to prove the convergence of the method and to obtain the same convergence rate as in the conforming case.
The mortar finite element method for Bingham fluids
This paper deals with the flow problem of a viscous plastic fluid in a cylindrical pipe. In order to approximate this problem governed by a variational inequality, we apply the nonconforming mortar finite element method. By using appropriate techniques, we are able to prove the convergence of the method and to obtain the same convergence rate as in the conforming case.
The Mortar method in the wavelet context
This paper deals with the use of wavelets in the framework of the Mortar method. We first review in an abstract framework the theory of the mortar method for non conforming domain decomposition, and point out some basic assumptions under which stability and convergence of such method can be proven. We study the application of the mortar method in the biorthogonal wavelet framework. In particular we define suitable multiplier spaces for imposing weak continuity. Unlike in the classical mortar method,...
The Mortar Method in the Wavelet Context
This paper deals with the use of wavelets in the framework of the Mortar method. We first review in an abstract framework the theory of the mortar method for non conforming domain decomposition, and point out some basic assumptions under which stability and convergence of such method can be proven. We study the application of the mortar method in the biorthogonal wavelet framework. In particular we define suitable multiplier spaces for imposing weak continuity. Unlike in the classical mortar method,...
Time domain decomposition in final value optimal control of the Maxwell system
We consider a boundary optimal control problem for the Maxwell system with a final value cost criterion. We introduce a time domain decomposition procedure for the corresponding optimality system which leads to a sequence of uncoupled optimality systems of local-in-time optimal control problems. In the limit full recovery of the coupling conditions is achieved, and, hence, the local solutions and controls converge to the global ones. The process is inherently parallel and is suitable for real-time...
Time Domain Decomposition in Final Value Optimal Control of the Maxwell System
We consider a boundary optimal control problem for the Maxwell system with a final value cost criterion. We introduce a time domain decomposition procedure for the corresponding optimality system which leads to a sequence of uncoupled optimality systems of local-in-time optimal control problems. In the limit full recovery of the coupling conditions is achieved, and, hence, the local solutions and controls converge to the global ones. The process is inherently parallel and is suitable for real-time...
Total overlapping Schwarz' preconditioners for elliptic problems
A variant of the Total Overlapping Schwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1 Math. 336 (2003) 277–282] as an iterative algorithm to approximate the absorbing boundary condition, in unbounded domains. That same method turns to be an efficient tool to make numerical zooms in regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture and the reliability to obtain the behavior of the solution...
Total overlapping Schwarz' preconditioners for elliptic problems
A variant of the Total Overlapping Schwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1 Math.336 (2003) 277–282] as an iterative algorithm to approximate the absorbing boundary condition, in unbounded domains. That same method turns to be an efficient tool to make numerical zooms in regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture and the reliability to obtain the...
Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra.
Two-grid finite-element schemes for the transient Navier-Stokes problem
We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size . In the second step, the problem is linearized by substituting into the non-linear term, the velocity computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size . This approach is motivated by the fact that,...
Two-grid finite-element schemes for the transient Navier-Stokes problem
We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size H. In the second step, the problem is linearized by substituting into the non-linear term, the velocity uH computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size h. This approach is motivated by the fact that,...
Two-level additive Schwarz preconditioners for fourth-order mixed methods.
Two-level nonlinear elimination based preconditioners for inexact Newton methods with application in shocked duct flow calculation.
Une méthode d'éléments finis hybrides en décomposition de domaines
Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation
For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal...
Uniform convergence of local multigrid methods for the time-harmonic Maxwell equation∗
For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal...
Use of the multigrid methods for heat radiation problem.