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Accueil > Agenda > Colloques et rencontres > Archives > Agenda 2002 > Workshop : Adaptative Methods for Evolution Problems

IRMA, 25-27 Novembre 2002

Scientific Committee :

W. Dahmen (Aachen)

D. Kroener (Freiburg)

E. Sonnendrücker (Strasbourg)

Local Organizers :

Michael Gutnic (gutnic@math.u-strasbg.fr)

Eric Sonnendrücker (sonnen@math.u-strasbg.fr)

Financial Support :

This workshop is supported by the ACI Jeunes Chercheurs "Analyse mathématique et simulation numérique de particules chargées" of the French Ministry of Research.

**PROGRAMME :**

**R. Becker** : * "Adaptive Finite Elements for Optimal Control".*

We present a systematic approach to error control and mesh adaptation for optimal control of systems governed by PDEs.

Starting from a coarse mesh, the finite element spaces are successively enriched in order to construct suitable discrete models. This process is guided by an a posteriori error estimator which employs sensitivity factors from the adjoint equation. We consider different examples involving the Navier-Stokes equations as the state equation. In addition to previous work [1-3], we address the question of error estimation of the control variable with respect to a natural norm.

References :

- [1] R. Becker : Mesh adaptation for stationary flow control, J. Math. Fluid Mech., 3(4):317—341, 2001.
- [2] R. Becker and R. Rannacher : A feed-back approach to error control in finite element methods : Basic analysis and examples, East-West J. Numer. Math., 4:237—264, 1996.
- [3] R. Becker and R. Rannacher : An optimal control approach to a-posteriori error estimation, In A. Iserles, editor, Acta Numerica 2001, pp. 1—102, Cambridge University Press, 2001.

**C. Bernardi** : *"Time and space adaptivity for parabolic problems".*

We are interested in the discretization of linear and nonlinear parabolic equations by an implicit scheme with respect to the time variable and finite elements with respect to the space variables. We propose two types of error indicators, with respect to both time and space approximations, and we prove their equivalence with the error, in order to work with adaptive time steps and finite element meshes.

** S. Bertoluzza and Y. Maday** : *"Analysis of a wavelet based adaptive scheme for the Burger’s equation".*

We consider and adaptive wavelet based algorithm for the solution of the viscous Burger’s equation with periodic boundary conditions in one dimension. In such an algorithm, based on an implicit/explicit Euler time discretization scheme, the space in which the solution is sought at each time step is designed by taking advantage of the information provided by the size of the wavelet coefficients of the solution at the previous time step. We provide an error estimate, under suitable assumptions on the parameters.

** M. Campos-Pinto** : *"A mesh refinement strategy and its error analysis for a fully adaptive linear scheme".*

We study the simple problem of transporting a univariate solution with a fully adaptive linear scheme, the solution being compressed in a wavelet basis. Using a decay property of the evolution operator associated to the upwind scheme, we show how to refine the adaptive mesh in order to guarantee that the error due to the adaptive scheme after n steps grows at most like n^(1/2).

**J. Cao** : * "Adaptive Finite Element Methods Coupled with A Posteriori Error Estimates - a Reliable Tool for Accurate Simulation of Engineering Problems".*

This talk focuses on applications of finite element mesh adaptivity oriented by a posteriori error estimation to computer simulations of Navier-Stokes flow and groundwater pollution remediation problems.

Linearized problems are first respectively generalized from the original nonlinear Navier-Stokes model and reactive advection-diffusion model. Then, the errors of finite element solution within each element on a mesh are estimated by solving a local Neumann problem corresponding to each linearized problem. The resulting a posteriori error estimates are further used to form both local and global energy norms serving as error index to be approximately equi-distributed over the mesh.

When meshes are deployed, triangles on the mesh are simultaneously refined and unrefined in a nested pattern, resulting in a hierarchical series of adaptive meshes ; the option of bisection-style refinement is also included that can lead to a triangulation with continuously-varying granularity. In addition to mesh refinement and/or unrefinement, nodes are also allowed to move on the mesh within a reasonably closed region to further improve the mesh quality.

Through various selected numerical experiments of aerodynamics and environmental engineering problems, the a posteriori error estimate methodology proves to be general in its approach while the different adaptive-gridding schemes serve as "black boxes". The coupling of these two novel finite element tools allows for an economical and accurate solution, and looks promising in solving with high efficiency more sophisticated problems in different engineering disciplines.

**A. Cohen** : *"Adaptive multiscale schemes for evolution equations".*

In this talk, we shall first review some approximation results which provide with a fundational ground for adaptive methods, either based on wavelets or mesh refinement. We then discuss a multiresolution processing strategy which applies to general classes of discretization schemes for evolution equations, based on a general discrete multiresolution framework proposed by Ami Harten. The goal of this processing is to reduce significantly the CPU cost and the memory space of the scheme, by making it adaptive, while preserving its order of accuracy. This strategy combines ideas of data compression by wavelet thresholding with the practice of adaptive mesh refinement.

**D. Goujot** : * "Hybrid N-term Approximation Methods For Control of Tree-Cardinality : a Step Towards Optimal Order Resolutions Schemes".*

The complexity of schemes often depends on the cardinality of the wavelet trees used to compute the solution. Here, the conventional N-term threshold does no longer apply : although it helps to ensure that the wavelet trees contain only N wavelet coefficients, it gives no indication of total size of these wavelet trees. The usual trick in this case is either to waste a small amount of regularity in the high order term, or to bound the maximal allowed refinement (e.g., when using wavelets to accelerate a resolution scheme on uniform grids). However, such tricks will probably not be allowed when conceiving faster resolution schemes.

Hence, the N-term threshold needs to be modified to give more direct control on the size of the wavelet tree. We present such a modification.

** F. Hecht, F. Alauzet, P.J. Frey and B. Mohammadi** : * "Metric mesh adaption for time-dependent problems".*

This presentation deals with the adaptation of meshes for transient problems. The proposed approach is based on a metric mesh adaptation algorithm and a metric intersection in time procedure suitable to capture such phenomena. More precisely, a new specific loop is inserted in the main adaptation loop to solve a transient fixed point problem. The mesh adaptation stage consists in optimizing the current mesh so as to obtain a unit mesh with respect to this metric. A 2D example with FreeFem++ [3] is provided to show the efficiency of the proposed method.

In the past, we have used these techniques for steady inviscid and viscous laminar configurations. Moreover, this method allows to substantially reduce the computational cost of the numerical simulation by reducing the mesh size. However, when dealing with unsteady configurations, it is of the utmost importance to follow the evolution of the physical phenomena (for instance a moving shock in the computational domain). It is thus necessary to mesh, in an adequate manner, all the regions where phenomena evolve. But attention must be paid so as to avoid remeshing a part too large of the domain with a dense mesh or remeshing too often thus impacting the overall computational cost.

We propose a mesh adaptation algorithm, based on the resolution of a time-dependent fixed point problem for the couple mesh-solution and likely suitable to predict the evolution of the physical phenomena. This algorithm is composed of two embedded loops. At each iteration of the main adaptation loop, a time period $[t,t+\Delta t]$ is considered where the solution evolves. In the inner loop, the transient fixed point problem is solved. At each internal iteration step (from $t$ to $t+\Delta t$), a new adapted mesh is created based on the metric associated with the solution at $t+\Delta t$ and the computation is re-started with the same initial solution at $t$. The inner process is repeated until the convergence ( i.e., the desired accuracy of the solution) is achieved at $t+\Delta t$. Then, we resume the outer adaptation loop at $t+\Delta t$ and the whole process is iterated [1]. Unstructured mesh adaptation is based on the computation of the edge lengths with respect to a discrete metric [2]. Specific algorithms have been designed so as to reduce the impact of mesh modifications at each mesh adaptation step (most of the current mesh entities are preserved in the areas where the solution is not changing). The anisotropic metric is defined using an a posteriori error estimate based on a discrete approximation of the Hessian of the solution. The aim is to equi-distribute the interpolation error along the mesh edges. In order to properly mesh all regions where the solution evolves, a metric intersection in time is introduced in the metric definition. In three dimensions specifically, this such-defined computational metric is intersected with the discrete geometric metric (based on the intrinsic properties of the surface) so as to preserve the domain geometry.

In this presentation, we will mainly focus on the metric construction and the adaptation scheme for unsteady problems. A 2D and 3D numerical example is provided so as to emphasize the efficiency of the proposed approach.

References :

- [1] F. Alauzet et al., 2002, Transient fixed point based unstructured mesh adaptation, Int. J. numer. methods fluids, to appear. Computational Field Simulations, Mississipi State Univ., 1996.
- [2] P.J. Frey et P.L. George, 1999, Maillages. Applications aux éléments finis, Hermès Science, Paris.
- [4] F. Hecht and B. Mohammadi, 1997, Mesh adaptation by metric control for multi-scale phenomena and turbulence, AIAA, paper 97-0859.
- 5] R. Loehner and J.D. Baum, 1992, Adaptive h-refinement on 3D unstructured grids for transient problems, Int. J. numer. methods fluids, Vol. 14, pp. 1407-1419.
- [6] B. Mohammadi, P. L. George, F. Hecht and E. Saltel, 2000, 3D Mesh adaptation by metric control for CFD, Revue Européene des Éléments Finis, Vol. 9, no. 4, pp. 439-449.
- [7] B. Mohammadi and F. Hecht, 2001, Mesh adaptation for time dependent simulation, optimization and control, Revue Européene des Éléments Finis, Vol. 10, no. 5, pp. 575-593.
- [8] R.D. Rausch, J.T. Batina and H.T.Y. Yang, 1992, Spatial adaptation procedures on tetrahedral meshes for unsteady aerodynamic flow calculations, AIAA J., 30, pp. 1243-1251.

[3] F.Hecht, O. Pironneau : freefem++ Manual, on the web at http://www-rocq.inria.fr/Frederic.Hecht/freefem++.htm

** V. Jovanovic** : * "A priori error estimates for Friedrichs’ system".*

We consider a class of finite-volume schemes on unstructured meshes for symmetric hyperbolic linear systems of balance laws in two space dimensions. This class of schemes has been introduced by Vila and Villedieu. They have proven an a-priori error estimate for approximations of smooth solutions. We extend the result to weak solutions. Furthermore an a-posteriori error estimate can be derived.

**J. Lang** : * "A Moving Finite Element Method with Local Refinement".*

We consider the numerical solution of time-dependent PDEs using a finite element method based upon rh-adaptivity. The quite robust h-method, where the mesh is locally refined or coarsened by adding or deleting mesh points, is combined with an r-method, which dynamically redistributes the mesh points in time. Moving mesh methods are superior at reducing dispersive errors in the vicinity of wave fronts while local refinement methos can, in principle, add enough degrees of freedom to resolve any fine structure. Numerical results are presented to demonstrate the capabilities and benefits of this approach.

**S. Müller** : *"Adaptive Multiscale Finite Volume Methods : Construction, Theory and Application".*

A new approach is presented by which any standard finite volume method can be accelerated. The basic idea is to incorporate data compression strategies based on wavelet techniques which has been originally suggested by Ami Harten.

Starting point is a so-called multiscale decomposition corresponding to a sequence of nested grids which is determined by the discrete flow field at hand. To this end, an array of cell averages corresponding to a finest resolution level is decomposed into to an equivalent array of cell averages on a coarsest resolution level and details describing the difference of the solution on two successive resolution levels. Since the details may become negligible small in regions where the flow field exhibits a moderate variation in the data, the complexity of the data can be reduced applying hard thresholding techniques to the multiscale decomposition. By means of the truncated sequence of multiscale coefficients a locally adapted grid with hanging nodes is predicted on which the time evolution is performed.

In collaboration with Albert Cohen et al. it has been possible to verify analytically that the threshold error introduced in each time step can be controlled provided the threshold value is judiciously chosen, i.e., the error does not blow up over all time steps. This result implies that the accuracy of the reference finite volume method can be maintained.

Moreover, numerical computations show that the resulting adaptive finite volume method is much more efficient than the reference finite volume method. In particular, the gain in computational time as well as the reduction of memory requirements are improved with an increasing number of refinement levels. This is different to Harten’s original concept which is only aiming at the reduction of expensive numerical flux computations, i.e., the complexity of the scheme still corresponds to that of the finest resolution level. Here the complexity is proportional to the number of significant details.

Besides constructional and analytical aspects of the adaptive scheme a special emphasis will be put on numerical computations. By means of several multidimensional test configurations we will verify that the new concept can be applied to real world problems such as steady state computations of flow fields around airfoils.

**M. Ohlberger** : * "Higher order finite volume methods on selfadaptive grids for convection dominated reactive transport problems in porous media. ".*

For advection equations it is well known that so called first order finite volume schemes are only of order $h^*1/2*$, if $h$ denotes the mesh size. In addition, those methods introduce some artificial numerical viscosity which is proportional to the local mesh size. Due to this drawbacks, such first order schemes on uniform computational grids are not very accurate. The situation gets even worse, if nonlinear reactions are also taken into account. In this situation the physical balance between reaction, advection and diffusion my be completely destroyed by the artificial numerical viscosity.

In order to cope with an accurate and efficient simulation of such flow phenomena we propose an higher order finite volume scheme on self-adaptive computational grids, where the adaptivity is steered by an rigorous a posteriori error estimate of a first order scheme.

The a posteriori error estimate is obtained for very general non-linear weakly coupled systems of convection-diffusion-reaction equations and especially hold true for vanishing diffusion coefficients.

We will demonstrate the efficiency of the resulting scheme in several numerical experiments concerning flow and transport in porous media.

**I. Paun** : * "An adaptive Vlasov-Poisson solver based on multi-resolution analysis".*

Simulation of some problems in plasma physics or for high intensity beams requires the numerical resolution of the Vlasov equation on a mesh of phase space which doubles the dimension which can be equal to as many as six for realistic problems. In order to optimize the number of mesh points where the distribution function is computed, we developed a Vlasov solver using a Multi Resolution Analysis. The distribution function is expanded on a wavelet basis spanning several scales. This allows us to discard the mesh points where the details are small. The idea of the algorithm is to couple this method for choosing the necessary grid points with a semi-Lagrangian type algorithm, where the characteristics are followed backwards from the selected mesh points, with time splitting between position and velocity advance. Our initial results using this method are presented.

** J. Proft** : * "Multi-algorithmic Adaptive Finite Element Numerical Strategies".*

We address the development of adaptive continuous and discontinuous finite element methods for simulating convective-diffusive flow models. Continuous finite element methods typically perform very efficiently and accurately on smooth flow problems but degrade in regions in which the solution is "rough" or advection is dominant. In contrast, discontinuous Galerkin methods typically perform well even in high flow gradient areas. Moreover, they have a number of important features rendering them useful for numerical computation. A multi-algorithmic approach to the numerical solution of convection-diffusion problems can couple and exploit the best features of both conforming/continuous and discontinuous Galerkin methods. We explore adaptive procedures for switching between methods based on a posteriori estimations.

**A. Rault, G. Chiavassa, R. Donat** : * "Shock-Vortex interactions at high Mach numbers".*

We perform a computational study of the interaction of a planar shock wave with a cylindrical vortex. We use a particularly robust High Resolution Shock Capturing scheme of third order coupled with multilevel technique based on Harten multiresolution. We obtain, at low numerical cost, high quality and high resolution numerical simulations of the interaction. In the case of a very-strong shock/vortex encounter, we observe a severe reorganization of the flow field in the downstream region, which seems to be due mainly to the strength of the shock. The numerical data is analyzed to study the driving mechanisms for the generation of acoustic waves and the production of vorticity in the interaction.

**W. Rosenbaum** : * "Adaptive staggered grids in 3D".*

Staggered grid schemes are qualified for the approximate solution of hyperbolic systems of conservation laws. The ease of evaluating fluxes not on the border but inside a cell, no requirement for (approximate) Riemann solvers and the componentwise application of the scalar framework to solve systems of conservation laws mak e these schemes very convenient to work with.

Besides these obvious advantages there are algorithmical difficulties caused by the use of staggered grids. One of these issues which is important for practical applications, is local adaptive grid refinement and coarsening, in particular for unsteady flows. Since the grids are now staggered, new techniques need to be developed. Here we attack the problem of local grid adaptation using structured staggered grids.

**O. Roussel** : * "An accurate and efficient adaptive multiresolution scheme for non-linear parabolic PDEs : application to combustion instabilities".*

The numerical simulation of combustion instabilities requires a large number of spatial and temporal scales. For a full resolution of the flame front, adaptive methods are highly recommended. Multiscale methods are therefore of particular interest due to their efficient data compression while controlling the local approximation error.

Here we present such an adaptive mutliresolution scheme for computing non-linear parabolic PDEs. The scheme is based on a classical finite volume discretization. We show its accuracy and efficiency for different test cases in one, two and three space dimensions. Then we apply the method to the simulation of spherical flame instabilities in 2D and 3D using the thermodiffusive approximation.

**T. Sonar** : * "Solution-adaptive dissipation models".*

In the 1940s von Neumann came up with the idea to stabilize a finite difference scheme for hyperbolic equations by means of an artificial diffusion. This idea succeeded in the famous gas dynamics codes of Jameson et al. in the 1980s when upwind, TVD- and ENO-strategies where introduced and built-in dissipation became fashion. Modern developments in image processing brought new ideas in the area of nonlinear diffusion equations and TV-Filters in the late 1990s. We took some of these ideas and developed new dissipation terms for finite difference schemes. On the basis of the Perona-Malik model we introduce solution-adaptive dissipation. In the case of TV filters we analyze some of the common models in image processing and show some numerical results.

**G. Warnecke** : * "Adaption and Stiffness of Computations for Evolution Problems".*

The talk will focus on local explicit-implicit switching of time-stepping methods in conjunction with local mesh adaption for partial differential

equations. A local $\Theta$-BDF method, partitioning as well as a local Krylov approach will be discussed. Computational examples for advection-diffusio and reaction-diffusion equations will be given

**U. Weikard** : * "Adaptive Numerics of the Cahn-Hilliard model".*

After introducing and motivating the Cahn-Hilliard model we present a fully discrete finite element approximation. We derive local a posteriori error estimates and show how use them for mesh adaption.

Furthermore, we consider a fully discrete approximation scheme for the Cahn-Hilliard equation with elasticity. In this model the elasticity tensor C depends on the concentration . Additionally anisotropic effects can be incorporated via an appropriate choice of C. Numerical calculations with anisotropic, inhomogenous elasticity yield results that closely resemble experimental data.

Dernière mise à jour le 17-10-2006