scholarly journals The Mixed Finite Element Multigrid Method for Stokes Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
K. Muzhinji ◽  
S. Shateyi ◽  
S. S. Motsa
Keyword(s):  
Finite Element ◽  
Multigrid Method ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Mixed Finite Element ◽  
Iterative Solvers ◽  
Algebraic Equations ◽  
Multigrid Solver ◽  
Element Discretization ◽  
Linear Algebraic Equations

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-TaylorQ2-Q1pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results.

Schwarz Type Solvers for -FEM Discretizations of Mixed Problems

2012 ◽  
Vol 12 (4) ◽  
pp. 369-390
Author(s):  
Sven Beuchler ◽  
Martin Purrucker
Keyword(s):  
Finite Element ◽  
Stokes Problem ◽  
Schur Complement ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Contraction Ratio ◽  
Linear Algebraic Equations ◽  
Stokes Problems ◽  
Elasticity Problems ◽  
Element Pair

AbstractThis paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development of an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair which satisfies the discrete inf-sup-condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast $hp$-FEM preconditioners for the potential equation. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressure is discretized by discontinuous finite elements. We will prove that the system of linear algebraic equations can be solved in almost optimal complexity. This yields quasioptimal hp-FEM solvers for the Stokes problems and the linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.

A TRUNCATED FOURIER/FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS IN AN AXISYMMETRIC DOMAIN

2006 ◽  
Vol 16 (02) ◽  
pp. 233-263 ◽  
Author(s):  
Z. BELHACHMI ◽  
C. BERNARDI ◽  
S. DEPARIS ◽  
F. HECHT
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Stokes Problem ◽  
A Priori ◽  
Three Dimensional ◽  
Angular Variable ◽  
A Posteriori ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Good Agreement

We consider the Stokes problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of this problem which combines Fourier truncation and finite element methods applied to each of the two-dimensional systems. We give the detailed a priori and a posteriori analyses of the discretization and present some numerical experiments which are in good agreement with the analysis.

Stabilized numerical method for solving the Oseen type problem with a singularity

2018 ◽  
Author(s):  
А.В. Рукавишников
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Krylov Subspace ◽  
Stokes Equations ◽  
Domain Decomposition Method ◽  
Algebraic Equations ◽  
Type Problem ◽  
Curvilinear Boundary ◽  
Linear Algebraic Equations ◽  
Element Method

На основе метода декомпозиции области построен стабилизационный неконформный метод конечных элементов для решения задачи типа Озеена. Для конвективно доминирующих течений с разрывным коэффициентом вязкости определена шкала оптимального выбора стабилизирующего параметра. Результаты численных экспериментов согласуются с теоретической оценкой сходимости. Purpose. To construct modified approximation approach using the finite element method and to perform numerical analysis for a two dimensional problem on the flow of a viscous inhomogeneous fluids — the Oseen type problem, that is obtained by sampling in time and linearizing the incompressible Navier—Stokes equations. To consider the convection dominated flow case. Methodology. Based on the domain decomposition method with a smooth curvilinear boundary between subdomains, a stabilization nonconformal finite element method is constructed that satisfies the inf-sup-stability condition. To solve the resulting system of linear algebraic equations, an iterative process is considered that uses the decomposition of the vector in the Krylov subspace with minimal inviscidity, with a block preconditioning of the matrix. Findings. The results of the numerical experiments demonstrate the robustness of the considered method for different (even small) discontinuous values of viscosity. The differences between finite element and exact solutions for the velocity field and pressure in the norms of the grid spaces decrease as

scholarly journals Finite element error analysis of surface Stokes equations in stream function formulation

2020 ◽  
Vol 54 (6) ◽  
pp. 2069-2097
Author(s):  
Philip Brandner ◽  
Arnold Reusken
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Stream Function ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Coupled System ◽  
Discretization Method ◽  
Element Discretization ◽  
Function Formulation ◽  
Stream Function Formulation

We consider a surface Stokes problem in stream function formulation on a simply connected oriented surface Γ ⊂ ℝ3 without boundary. This formulation leads to a coupled system of two second order scalar surface partial differential equations (for the stream function and an auxiliary variable). To this coupled system a trace finite element discretization method is applied. The main topic of the paper is an error analysis of this discretization method, resulting in optimal order discretization error bounds. The analysis applies to the surface finite element method of Dziuk–Elliott, too. We also investigate methods for reconstructing velocity and pressure from the stream function approximation. Results of numerical experiments are included.

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