Variational Multiscale Methods For The Incompressible Navier-Stokes Equations

In recent half century, along with the rapid development of computer science, numericalanalysis and numerical computation have been developed as very important tools in thefelds of science researches and engineering applications, and fnally formulated a newinterdisciplinary—Computational Fluid Dynamics(CFD). The incompressible Navier-Stokes equations (NSEs) are a kind of important problems in CFD, and are typicalnonlinear equations which widely used in the feld of science and engineering. However,due to their nonlinearity, especially, with high Reynolds number, it’s difcult to makemathematic analysis and numerical computation.For NSEs with high Reynolds number, Direct Numerical Simulation(DNS) usually re-quires very fne mesh size (h≈Re3/4,3D), huge storage space and much more CPU timeto capture the very small information. Thus, DNS for solving NSEs is still a big challengeeven in the highly-developed computer science nowadays. On the other hand, in manyliteratures, the fow is usually considered as consisting of three scales: the large scales,the resolved small scales and the unresolved small scales. Generally, based on suitablemesh size(h Re3/4), the classical fnite element spaces can seize the large scales andthe resolved small scales. The way to model the infuence of the unresolved small scalesonto the resolved small scales is the key to construct the efective numerical schemes forNSEs with high Reynolds number. As we known, Variational Multiscale(VMS) methodsregard the standard fnite element space as the resolved scales space, and its appropriateprojection space as the large scales space, then simulate the infuence of the unresolvedsmall scales to the resolved small scales through some formulations based on the resolvedsmall scales. Herein, we make some improvements and developments based on the VMSmethods as follows: First of all, for the steady NSEs, the convergence of a general algorithm with Newton it-eration for the VMS method is presented. Meanwhile, two special algorithms of explicit-and implicit-types with linear-and quadratical-convergence are derived from the generalalgorithm, respectively. Finally, some numerical analysis and tests of these two algo-rithms on the convergence conditions and orders are shown.Secondly, a fnite element VMS method for incompressible fows based on two localGauss integrations (GVMS) is presented. Compared with the classical VMS method(CVMS), based on the Taylor-Hood element, the best algorithmic feature of GVMS isusing two local Gauss integrations to avoid the construction of projection from the re-solved scales space onto the resolved small scales space, and adding the stabilization atthe element level without introducing any new variables or storage space. Meanwhile, themethod keeps the stability and efectiveness of CVMS. Finally, we theoretically discussthe equivalence and diferences between GVMS and CVMS in view of matrix analysisand numerical tests.Then, we consider VMS method with h-adaptive technique for the stationary NSEs.The natural combination of VMS with adaptive strategy retains the best features ofboth methods and conquers many of their disadvantages. Based on the ideas of VMSmethod and the equivalence between the unresolved small scales and the approximationerror, under the assumption of the unresolved small scales controlled by the resolvedsmall scales, we derive a general a posteriori projection error estimator based on theresolved small scales. Especially, a reliable a posteriori error estimator using two localGauss integrations at the element level and related adaptive GVMS are derived.Finally, a VMS method for incompressible fows based on the projection basis functions(VMSPBF) is presented. The attractive feature is constructing the projection basis func-tions at the element level with minimal additional cost to replace the global projectionoperator, and keeping the stability and efectiveness of CVMS. Moreover, neither sourceterms change nor Newton iterations proceed, one can pre-compute the projection basisfunctions once and re-use it. Since the construction of the projection basis functionsis independent of each other, they can be carried out in parallel perfectly. By adapt-ing the triangulations for the subproblems, we also can control the artifcial dissipation. Therefore, we present a VMS method based on adaptive projection basis functions (VM-SPBFAdap). Using the a posteriori projection estimator based on the resolved smallscales, appropriately choosing the parameters, VMSPBFAdap constructs the projectionbasis functions in a reasonable way.