Welcome to the TMR Project "Wavelets in Numerical Simulation"

Project Objectives

( Extract from TMR network Proposal "Wavelets and Multiscale Methods in Numerical Analysis and Simulation" )

The remarks of the previous section reflect the necessity of developping collaborations (and intensify those that already exist) between the nine research groups from six different european countries, with the ultimate common goal of applying wavelet-based methods to complex industrial problems. The realization of this goal requires dealing with the following main objectives : - A sound theoretical foundation addressing the following issues (1) so far methods unfold their full computational potential for problems on simple domains like squares. An issue of central importance is therefore the adaptation of multiscale decomposition to general domains and manifolds. Possible envisaged strategies include embedding techniques, domain- adapted wavelets and domain decomposition. (2) preconditioning elliptic problems is meanwhile understood. Extending such techniques to singularity perturbed or highly anisotropic, even non-elliptic problems have to be addressed. (3) the efficient evaluation of nonlinear terms applied to functions in multiscale form, which arises e.g. in the discretization of nonlinear PDEs, poses severe difficulties. The analysis and numerical approximation of such nonlinear terms is of predominant importance. (4) the discretization of integral operators, which often provide the physically more adequate mathematical model, usually leads to densely populated systems of equations. The availability of (nearly) sparse wavelet representations, their adaptive potential combined with the advantage of preconditioning is particulary promising. (5) we will study the coupling of multiscale methods with finite element methods. (6) the potential of adaotive techniques is closely related to the regularity of the approximated object relative to a scale of Besov spaces which in turn can be characterized through the concept of nonlinear approximation. The interrelation of these concepts has to be investigated for solutions of PDEs. - The insight gained by the theoretical investigations will identify the concrete, problem-dependant demands on the tools, and support the development of multiscale numerical schemes that combine high order approximation and adaptivity with respect to data, operators and solutions. These schemes will be flexible enough to be adapted to a variety of complex problems in the area of fluid dynamics, chemical engineering, electromagnetism and elasticity. They will be both based on wavelets and, if more appropriate, on multiscale decomposition strategies in the settings of more classical discretizations methods such as finite elements. In this context we shall also explore to what extend classical methods might benefit from the new ingredients emerging from this research. - A central objective is the development of codes and software based on these schemes for the numerical solution of industrial problems and their optimization, iin order to improve the performance of existing software which is based on more classical methods. More precisely, we plan to open the last meeting of the research program to a large number of participants, including experts in the application of finite element methods. This final meeting will be mainly devoted to comparisons of conventional and new schemes for relevant test problems.