Schwache Form - Herleitung durch Partielles Integrieren - Finite Elemente

The weak form reduces the continuity requirements of the basis functions used for approximation, allowing the use of lower-degree polynomials. This is achieved by transforming the differential equation into an integral form, which is generally easier to solve. The weak formulation is one of the reasons for the widely known technique of increasing the number of elements for higher simulation accuracy. A common FEM example problem to illustrate this is: Consider the stresses on a free surface. According to the strong form, the stresses should be zero when no force is applied to the surface. Solving the PDEs should yield stress values ​​of zero. However, commercial FE codes report some value for the surface stresses because they are all based on weak formulations, which rely on integral formulas. Therefore, increasing the mesh size in this region tends to reduce the stresses to zero. This illustrates that accurate solutions can be obtained in a weaker form with a higher mesh density. Using higher-order form functions is another method for improving accuracy, in addition to employing efficient stabilization procedures. Once the weaker integral formulation is obtained, it can be transformed into an (algebraic) matrix formulation, which is easier to solve because many well-established and tested algorithms exist for this purpose. Translated with www.DeepL.com/Translator (free version) Integration by parts (Latin: integratio per partes), also called integration by products, is a method in integral calculus for calculating definite integrals and determining antiderivatives. It can be considered analogous to the product rule in differential calculus. Gauss's theorem from vector analysis, along with some of its special cases, is a generalization of integration by parts for functions of several variables.