Scalable Parallel Computational Methods for Partial Differential Equations with Moving and Varying Boundaries

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This project will investigate novel parallel numerical methods, relying on a special type of finite element meshes, called here LRLFH-meshes (Locally Refined Locally Fitted Hierarchical meshes), and on coupling of advanced multigrid methods with domain embedding/fictitious domains and domain decomposition techniques, for solving linear and nonlinear Partial Differential Equations with moving and varying boundaries. The efficient solution of such problems is very important in various scientific and industrial applications, e.g., particulate flows, optimal control of systems with moving parts, optimal shape design. In the case of problems with varying geometry, requiring the solution of a large number of finite element/finite volume systems of equations associated with varying meshes, finding the right balance between accuracy and algorithmic efficiency is very difficult, particularly on parallel platforms. Recent results indicate that highly efficient parallel methods can be designed for the above problems by coupling multigrid preconditioning with domain decomposition and fictitious/embedding domain techniques on meshes which are block structured or almost structured, respectively. The main objectives of this project are:To develop and investigate parallel generators for Locally Refined Locally Fitted Hierarchical meshes for complex two and three dimensional geometries.To develop and investigate scalable parallel iterative solution methods with preconditioning based on coupling advanced multigrid algorithms with fictitious (embedding) domain and domain decomposition techniques.To apply the mesh generators and iterative solution methods, to be developed, to the numerical simulation of flow problems with complex moving boundaries, e. g., particulate flows governed by the Navier-Stokes equations, and to optimal shape design problems on parallel platforms (clusters of workstations and IBM SP/2).The work will have impact on the simulation of various phenomena encountered in chemical, mechanical and aerospace engineering.