Fast Spectral Element Methods For Wave Equation And Navier-Stokes Equations And Its Applications

The spectral element method takes advantage of the high-order precision of thespectral methods and the geometrical ?exibility of the finite element methods. Itbecomes one of the most popular methods in numerically solving PDE problems.However it is known that the computational cost of the spectral element methodis higher than other lower order methods when the number of grid points is same.Therefore there is a need for the development of fast solvers of the spectral elementmethod.This paper consists of four parts. In the first part, we attempt to provide a detaileddiscussion on the temporal discretizations of the Navier-Stokes equations currentlyemployed in the computation of incompressible ?ows. Especially, we will introduce asimple PN×PN method with no need to use staggered grid. The method is based onthe projection schemes, together with a PN→PN?2 post-filtering. Numerical testsshow that this PN×PN method removes the numerical pressure oscillations by thetraditional PN×PN projection methods when very small time steps are used.The second part of this paper contributes to construct a new fast solver based onthe Schur complement algorithm to solve both velocity and pressure systems withinthe frame of PN×PN spectral element spatial discretization. We will provide a morecomprehensive description of the Schur complement method by using nodal basis,and suggest an implementation strategy to reduce the computational cost. In thisdescription, the Schur complement system, involving the interface unknowns, is writtenas a sum of the local (elemental) Schur complement systems, which stem from thelocal problems with Neumann boundary conditions. The implementation strategy wesuggest consists of- in 2D case, saving the local Schur complement matrices for use of matrix-vectorproducts when an iterative method such as conjugate gradient method is applied.- in 3D case, using a reduced expression of the Schur complement system toevaluate the matrix-vector products if there are deformed macro-elements.A detailed comparison in terms of computational and storage cost is carried outfor the above two di?erent ways of evaluating the matrix-vector multiplications. Moreover, we propose a fast diagonalization method (FDM) to solve the localsubproblems. Precisely, we propose e?cient preconditioners for the individual localsubproblems defined in non-rectangular domain, and these preconditioners are con-structed by applying FDM to appropriate elliptic problems defined in the referencerectangular domain.In the third part we apply the above methods, together with a new LES model,to the large eddy simulations of 3D turbulence. The new LES model is the so-calledspectral vanishing viscosity model, which is based on the original NS equations and iseasy to implement without significant increase of the computational cost. We simulatethe lid-driven cubic cavity ?ows at Reynolds number Re = 12000, the results show agood agreement with existing experimental and DNS results found in the literature.Thanks to the fast solver developed in this paper, we are able to carry out the abovesimulation on a personal computer.In the fourth part, we consider the approximation of acoustic wave propagationproblems by the Newmark schema in time and spectral element method in space.Some detailed stability and error estimates are obtained. From these results, thespectral accuracy and in?uences of the non-homogeneous boundary data are madeevident. Several numerical examples are provided to confirm the theoretical analysis.The advantage of the present method is demonstrated by a numerical comparisonwith the finite element method.