Partition of Unity Methods

1 Introduction

Stéphane P. A. Bordas¹, Alexander Menk², and Sundararajan Natarajan³

¹ University of Luxembourg, Luxembourg, UK² Robert Bosch GmbH, Germany³ Indian Institute of Technology Madras, India

Physical systems are often modeled using partial differential equations (PDEs). The exact solution or closed form or analytical solutions to these PDEs is only available in special cases for specific geometries. Numerical methods can be used to approximate the exact solution in more general settings. The result of a numerical simulation is rarely exact. Nonetheless, computer-based numerical simulation has revolutionalized industrial product development throughout engineering disciplines. When comparing experiments and simulation with the aim of improving a simulation procedure to give more accurate results, it is necessary to understand the different sources of error. Figure 1.1 shows an overview of errors that occur at different stages of modeling and numerical simulation for a given numerical method.

Figure 1.1 Sources of error in simulation.

One of these numerical methods is called the “finite element method” (FEM). It is most commonly used in structural mechanics, although the field of application is much broader. The historic origins of the FEM cannot be uniquely determined. Mathematicians and engineers seemed to develop similar methods simultaneously which laid the foundations ...