An element formulation for the classical finite difference and finite volume method applied to arbitrarily shaped domains

Abstract

AbstractPotential problems can be solved by analytical methods. However, for three‐dimensional structures of geometrically complex shape with non‐linear boundary conditions, numerical methods have to be applied. The most popular technique is the finite element method, which is used throughout the engineering community. The main advantage of this method lies in the application to arbitrarily shaped structures, which requires the use of non‐orthogonal meshes. More recently, the boundary element method has been proven to be a competitive tool in the field of potential problems.In other research areas different numerical methods are used, for example in fluid mechanics the finite volume or the finite difference methods. But the application to irregularly shaped domains is difficult compared to the finite element and boundary element methods. The big drawback of these methods is not a poor numerical quality, but the absence of an element concept. In some cases, the results are even better compared to the other methods. Therefore, the main subject of this paper is the introduction of the element concept not only for the popular finite element method, but also for the finite difference and finite volume methods, which provides a high degree of geometric flexibility.