Trefftz

In computational mechanics, a Trefftz method is a type of numerical method used to solve partial differential equations. These methods are characterized by using solution spaces spanned by functions that satisfy the governing differential equation exactly within the domain of interest. This is in contrast to methods like the finite element method, where the basis functions only satisfy the governing equation in a weak sense or through discretization.

The core idea behind Trefftz methods is to represent the solution as a linear combination of these "Trefftz functions". The coefficients of the linear combination are then determined by enforcing certain boundary conditions or interface conditions. Different Trefftz methods may vary in how they enforce these conditions, such as using collocation, least squares, or variational formulations.

Because the basis functions satisfy the governing equation exactly within the domain, Trefftz methods often lead to higher accuracy with fewer degrees of freedom compared to other methods, especially for problems with smooth solutions. However, constructing suitable Trefftz functions can be challenging, particularly for complex geometries or non-homogeneous materials.

Some common types of Trefftz methods include:

• Indirect Boundary Element Method (IBEM): A type of Trefftz method where the solution is represented by layer potentials defined on the boundary.
• Method of Fundamental Solutions (MFS): Utilizes fundamental solutions of the governing equation as basis functions, with source points located outside the domain.
• Element-Free Trefftz Methods (EF-TFM): Combines the advantages of Trefftz methods with element-free approximations, eliminating the need for mesh generation.

The name "Trefftz" honors the German mathematician Erich Trefftz, who pioneered these types of methods.