Recently, stable meshfree methods for computational fluid mechanics have attracted rising interest. So far such methods mostly resort to similar strategies as already used for stabilized finite element formulations. In this study, we introduce an information theoretical interpretation of Petrov-Galerkin methods and Green's functions. As a consequence of such an interpretation, we establish a new class of methods, the so-called information flux methods. These schemes may not be considered as stabilized methods, but rather as methods which are stable by their very nature. Using the example of convection-diffusion problems, we demonstrate these methods' excellent stability and accuracy, both in one and higher dimensions.
SUMMARYThe requirement for stabilization or other similar techniques is well known when using the finite element method in computational fluid mechanics. A variety of such techniques has been introduced during the past decades along with different physical interpretations of the stabilization terms employed. In introducing so-called information flux methods, we developed a new point of view on the problem of numerical instabilities; with respect to Shannon's information theory instabilities are interpreted as a consequence of unadequate observance of the information flux present in fluid mechanics. Here we discuss different approaches to setting up information flux maximum-entropy approximation schemes based on that idea. The good accuracy of these approximation schemes is demonstrated for convection-diffusion problems by means of several linear, time-independent one-and two-dimensional numerical examples and comparisons with state-of-the-art stabilized finite element methods.
SUMMARYWe present a parameter-free stable maximum-entropy method for incompressible Stokes flow. Derived from a least-biased optimization inspired by information theory, the meshfree maximum-entropy method appears as an interesting alternative to classical approximation schemes like the finite element method. Especially compared with other meshfree methods, e.g. the moving least-squares method, it allows for a straightforward imposition of boundary conditions. However, no Eulerian approach has yet been presented for real incompressible flow, encountering the convective and pressure instabilities. In this paper, we exclusively address the pressure instabilities caused by the mixed velocity-pressure formulation of incompressible Stokes flow. In a preparatory discussion, existing stable and stabilized methods are investigated and evaluated. This is used to develop different approaches towards a stable maximum-entropy formulation. We show results for two analytical tests, including a presentation of the convergence behavior. As a typical benchmark problem, results are also shown for the leaky lid-driven cavity. The already presented information-flux method for convection-dominated problems in mind, we see this as the last step towards a maximum-entropy method capable of simulating full incompressible flow problems.
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