Performance aspects of a mixed s‐v LSFEM for the incompressible Navier‐Stokes equations with improved mass conservation

Abstract

In the present contribution we propose an improved mixed least-squares finite element method (LSFEM) and compare it with standard LSFEMs with respect to performance aspects. In detail, we consider an approach for Newtonian fluid flow, which is described by the incompressible Navier-Stokes equations. The basis for the associated symmetric minimization problem is a reduced stress-velocity (s-v) two-field approach, see e.g. CAI ET AL.[1] and SCHWARZ & SCHRÖDER [2]. The main idea for the proposed formulation is to add an additional equation, which yields an overconstrained first-order system. This approach does not introduce additional unknowns, so the advantage of two variables (stresses and velocities) remains. Finally, we present a numerical example in order to show the capability of the proposed formulation.

Implicit time discretization schemes for least‐squares finite element formulations to model incompressible flows

“…In order to derive a first-order system we consider the constitutive law σ = 2ρν∇ s v − p1, where σ denotes the Cauchy stresses. The resulting nonlinear equation system in terms of stresses and velocities is reformulated, see [1] for the steady case, and consists of the three residuals…”

Investigation of stress‐velocity LSFEMs for the incompressible Navier‐Stokes equations

“…The extended formulation implies that a redundant residual is added to the functional without additional variables. This formulation is based on the deliberations in [1], the extended version can be found in [5] and [3].…”

“…The first formulation under consideration is the stress-velocity-pressure formulation according to [1]. Secondly, an extended stressvelocity formulation with an additional residual is derived based on the findings in [1] and [5]. The third formulation is a pressure reduced stress-velocity formulation based on a condensation scheme.…”

A fluid‐structure interaction approach with inherently fulfilled coupling conditions

“…Furthermore, the construction of the finite elements leads to positive definite and symmetric system matrices, also for differential equations with non self-adjoint operators and is applied successfully especially in fluid mechanics, see e.g. [1,4,5]. Besides, the LSFEM offers a wide range of approaches based on different solution variables, since they can be included directly in the formulations, which is a main point in the proposed approach.…”