3D hp-Finite Elements: Fast Solvers and Adaptivity

Mechanical deformations, magnetic fields, and many other applications from science and engineering are described by partial differential equations. The Finite Element Method (FEM) is the most powerful technique for computer simulation of these models. Here, the solution is approximated in a finite dimensional function space defined by means of a mesh. By increasing the space dimension, convergence of the FE approximation to the true solution is achieved. Since the beginnings of the FEM in the 50s, the accuracy is primarily increased by decreasing the mesh size. This is called h-version FEM. While rough functions can be best approximated on fine meshes, smooth functions can be approximated very well by high order polynomials on coarse meshes. Increasing the polynomial degree p while keeping the mesh fixed leads to the p-version FEM. In real world applications, solutions are neither everywhere smooth or rough, but have variable regularity over the domain. A methodology to combine the advantages of both in an optimal way is called hp-version. While h and p versions are limited to algebraic rates of convergence (in terms of the space dimension), the hp-version has an exponentially fast rate of convergence for a wide class of realistic problems. 2D Problems can be solved successfully by the hp-FEM. The mesh-refinement as well as the distribution of the polynomial degree can be chosen by a-priori knowledge, and for solving the arising system of linear equations direct elimination methods are appropriate. The 3D case is still a big challenge. One fundamental difference is the existence of edges in 3D domains. Thus, additionally to corner singularities already present in 2D, also edge singularities appear in the solution. These functions have anisotropic behaviour, and must be approximated on anisotropic meshes. The second difference is that the problem size in 3D is much larger in comparison to 2D. Thus, the selection of the hp-mesh decides if the problem fits into the available computer, or not. It is highly desired to have automatic hp- mesh refinement strategies which are based on the solution itself, i.e. a posteriori. After assembling the system of equations, solving is the next challenge. Direct elimination is out of the game for large problems, and iterative methods have to be applied. The speed of these methods heavily depends on the chosen preconditioner. The design of cheap and robust preconditioners is an open problem. The big challenge in the development of hp-FEM simulation software is the need of mathematical understanding in combination with skills in non-trivial code development. The proposer wants to take this challenge. He wants to develop an adaptive 3D hp-FEM software including fast iterative solvers.

Mechanical deformations in a machine frame, electromagnetic fields in a transformer, and many other applications from science and engineering can be mathematically modeled by so called partial differential equations. In this start project, we worked on the development, mathematical analysis and implementation of efficient algorithms for computer simulation of such models. By means of such simulation tools, the design engineer can accurately analyze his products during the development phase, and can optimize for material costs, energy efficiency, safety and lifetime, and other criteria. One of the application areas in the start project was mechanical and civil engineering. Here, one asks for the internal reaction forces of some structure due to external loads, for example the snow load on a roof. On one side, the weight minimization is an issue, on the other side, the structure must not break in the worst case situation. The other major application area was electrical engineering. Electric currents interact with magnetic fields, and lead to energy losses due to thermal heating. The common task in these applications is that one has to compute distributed field quantities. The finite element method is the most flexible and powerful method for such applications. Here, the fields are represented on a grid covering the domain of interest. Depending on the physics, the field values are represented by point values, mean values on cells, or fluxes. To achieve high accuracy, one either has to use very fine grids, or use smart (high order) methods for representing the fields. The hp-Version of the finite element method aims to combine both to obtain an optimal accuracy - cost ratio. In the start project we have developed new hp finite element methods for modeling electromagnetic and mechanical fields. The fields are described by many parameters (millions and more), which have to be found by solving large systems of equations. This is typically the most expensive part of the simulation. We have developed optimal iterative solvers for hp finite elements. In particular for electromagnetic fields, we have developed new mathematical tools for analyzing such solvers. Finally, the quality of the computed solution should be guaranteed by an error estimate. Here, we have found new mathematical techniques to deliver rigorous and sharp error bounds. A lot of effort was spent in the efficient software implementation of the mathematical algorithms. The resulting computer programs Netgen and NGSolve are available as open source software, and are used now by many teams in academia as well as in industry.