Newton multigrid least-squares FEM for the V-V-P formulation of the Navier–Stokes equations

“…Finally, after 2 years, in 1964, he presented a multigrid algorithm [33]. Then other researchers were interested in this issue, and much more research works appeared in the literature [34][35][36][37][38][39][40][41][42][43][44][45][46]. Donatelli et al [47][48][49][50] proposed different multigrid methods for linear systems, and then an analysis was presented for their methods.…”

Section: Introductionmentioning

confidence: 99%

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

Moghaderi

Dehghan

2016

Math Methods in App Sciences

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The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.

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“…They demonstrated superior convergence of the multigrid solver compared to the Jacobi preconditioning. Most recently, Nickaeen et al [2014] have developed a geometric multigrid solver as a preconditioner for the CG (MPCG) iterations to solve the vorticity formulation of the NS equations with LSFEM. A robust and grid independent behavior is demonstrated for the solution of different flow problems with both bilinear and biquadratic finite elements.…”

Section: Introductionmentioning

confidence: 99%

A comparative study of mixed least-squares FEMs for the incompressible Navier-Stokes equations

Schwarz

Nickaeen

Serdas

et al. 2016

IJCSE

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In the present contribution we compare different mixed least-squares finite element formulations (LSFEMs) with respect to computational costs and accuracy. In detail, we consider an approach for Newtonian fluid flow, which is described by the incompressible Navier-Stokes equations. Starting from the residual forms of the equilibrium equation and the continuity condition, various first-order systems are derived. From these systems least-squares functionals are constructed by means of L 2-norms, which are the basis for the associated minimization problems. The first formulation under consideration is a div-grad first-order system resulting in a threefield formulation with stresses, velocities, and pressure as unknowns. This S-V-P formulation is approximated in H(div) × H 1 × L 2 on triangles and for comparison also in H 1 × H 1 × L 2 on quadrilaterals. The second formulation is the well-known div-curl-grad first-order velocityvorticity-pressure (V-V-P) formulation. Here all unknowns are approximated in H 1 on quadrilaterals. Besides some numerical advantages, as e.g. an inherent symmetric structure of the system of equations and a directly available error estimator, it is known that least-squares methods have also a drawback concerning mass conservation, especially when lower-order elements are used. Therefore, the main focus of the work is on performance and accuracy aspects on the one side for finite elements with different interpolation orders and on the other side on the usage of efficient solvers, for instance of Krylov-space or multigrid type. In order to demonstrate the capability of the formulations the results for some well-known benchmark problems are presented and discussed.

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Newton multigrid least-squares FEM for the V-V-P formulation of the Navier–Stokes equations

Cited by 18 publications

References 23 publications

On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals

On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

A comparative study of mixed least-squares FEMs for the incompressible Navier-Stokes equations

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