8 research outputs found
Smoothing analysis of two-color distributive relaxation for solving 2D Stokes flow by multigrid method
Smoothing properties of two-color distributive relaxation for solving a
two-dimensional (2D) Stokes flow by multigrid method are theoretically
investigated by using the local Fourier analysis (LFA) method. The governing
equation of the 2D Stokes flow in consideration is discretized with the
non-staggered grid and an added pressure stabilization term with stabilized
parameters to be determined is introduced into the discretization system in
order to enhance the smoothing effectiveness in the analysis. So, an important
problem caused by the added pressure stabilization term is how to determine a
suitable zone of parameters in the added term. To that end, theoretically, a
two-color distributive relaxation, developed on the two-color Jacobi point
relaxation, is established for the 2D Stokes flow. Firstly, a mathematical
constitution based on the Fourier modes with various frequency components is
constructed as a base of the two-color smoothing analysis, in which the related
Fourier representation is presented by the form of two-color Jacobi point
relaxation. Then, an optimal one-stage relaxation parameter and related
smoothing factor for the two-color distributive relaxation are applied to the
discretization system, and an analytical expression of the parameter zone on
the added pressure stabilization term is established by LFA. The obtained
analytical results show that numerical schemes for solving 2D Stokes flow by
multigrid method on the two-color distributive relaxation have a specific
convergence zone on the parameters of the added pressure stabilization term,
and the property of convergence is independent of mesh size, but depends on the
parameters of the pressure stabilization term
A local Fourier analysis of additive Vanka relaxation for the Stokes equations
Multigrid methods are popular solution algorithms for many discretized PDEs,
either as standalone iterative solvers or as preconditioners, due to their high
efficiency. However, the choice and optimization of multigrid components such
as relaxation schemes and grid-transfer operators is crucial to the design of
optimally efficient algorithms. It is well--known that local Fourier analysis
(LFA) is a useful tool to predict and analyze the performance of these
components. In this paper, we develop a local Fourier analysis of monolithic
multigrid methods based on additive Vanka relaxation schemes for mixed
finite-element discretizations of the Stokes equations. The analysis offers
insight into the choice of "patches" for the Vanka relaxation, revealing that
smaller patches offer more effective convergence per floating point operation.
Parameters that minimize the two-grid convergence factor are proposed and
numerical experiments are presented to validate the LFA predictions.Comment: 30 pages, 12 figures. Add new sections: multiplicative Vanka results
and sensitivity of convergence factors to mesh distortio
Π ΠΊΠΎΠΌΠ±ΠΈΠ½ΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΡΠ°Π·Π»ΠΈΡΠ½ΡΡ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΡΡΠΊΠΎΡΠ΅Π½ΠΈΡ ΠΏΡΠΈ ΠΈΡΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΠΎΠΌ ΡΠ΅ΡΠ΅Π½ΠΈΠΈ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ Ρ ΡΠ°ΡΡΠ½ΡΠΌΠΈ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄Π½ΡΠΌΠΈ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠΌ ΠΊΠΎΠ»Π»ΠΎΠΊΠ°ΡΠΈΠΉ ΠΈ Π½Π°ΠΈΠΌΠ΅Π½ΡΡΠΈΡ Π½Π΅Π²ΡΠ·ΠΎΠΊ
In the work, we consider the problem of accelerating the iteration process of the numerical solution of boundary-value problems for partial differential equations (PDE) by the method of collocations and least residuals (CLR). To solve this problem, it is proposed to combine simultaneously three techniques of the iteration process acceleration: the preconditioner, the multigrid algorithm, and the correction of the PDE solution at the intermediate iterations in the Krylov subspace. The influence of all three techniques of the iteration acceleration was investigated both individually for each technique and at their combination. Each of the above techniques is shown to make its contribution to the quantitative figure of iteration process speed-up. The algorithm which employs the Krylov subspaces makes the most significant contribution. The joint simultaneous application of all three techniques for accelerating the iterative solution of specific boundary-value problems enabled a reduction of the CPU time of their solution on computer by a factor of up to 230 in comparison with the case when no acceleration techniques were applied. A two-parameter preconditioner was investigated. It is proposed to find the optimal values of its parameters by the numerical solution of a computationally inexpensive problem of minimizing the condition number of the system of linear algebraic equations. The problem is solved by the CLR method and it is modified by the preconditioner. It is shown that it is sufficient to restrict oneself in the multigrid version of the CLR method only to a simple solution prolongation operation on the multigrid complex to reduce substantially the CPU time of the boundary-value problem solution.Numerous computational examples are presented, which demonstrate the efficiency of the approaches proposed for accelerating the iterative processes of the numerical solution of the boundary-value problems for the two-dimensional NavierβStokes equations. It is pointed out that the proposed combination of the techniques for accelerating the iteration processes may be also implemented within the framework of other numerical techniques for the solution of PDEs.Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΡΡΠΊΠΎΡΠ΅Π½ΠΈΡ ΠΈΡΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠΈΡΠ»Π΅Π½Π½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠΌ ΠΊΠΎΠ»Π»ΠΎΠΊΠ°ΡΠΈΠΉ ΠΈ Π½Π°ΠΈΠΌΠ΅Π½ΡΡΠΈΡ
Π½Π΅Π²ΡΠ·ΠΎΠΊ (ΠΠΠ) ΠΊΡΠ°Π΅Π²ΡΡ
Π·Π°Π΄Π°Ρ Π΄Π»Ρ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ Ρ ΡΠ°ΡΡΠ½ΡΠΌΠΈ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄Π½ΡΠΌΠΈ (PDE). ΠΠ»Ρ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΡΡΠΎΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ Π²ΠΏΠ΅ΡΠ²ΡΠ΅ ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½ΠΎ ΠΊΠΎΠΌΠ±ΠΈΠ½ΠΈΡΠΎΠ²Π°Π½Π½ΠΎ ΠΏΡΠΈΠΌΠ΅- Π½ΡΡΡ ΠΎΠ΄Π½ΠΎΠ²ΡΠ΅ΠΌΠ΅Π½Π½ΠΎ ΡΡΠΈ ΡΠΏΠΎΡΠΎΠ±Π° ΡΡΠΊΠΎΡΠ΅Π½ΠΈΡ ΠΈΡΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ ΠΏΡΠΎΡΠ΅ΡΡΠ°: ΠΏΡΠ΅Π΄ΠΎΠ±ΡΡΠ»Π°Π²Π»ΠΈΠ²Π°ΡΠ΅Π»Ρ, ΠΌΠ½ΠΎΠ³ΠΎΡΠ΅ΡΠΎΡΠ½ΡΠΉ Π°Π»Π³ΠΎΡΠΈΡΠΌ ΠΈ ΠΊΠΎΡΡΠ΅ΠΊΡΠΈΡ ΡΠ΅ΡΠ΅Π½ΠΈΡ PDE Π½Π° ΠΏΡΠΎΠΌΠ΅ΠΆΡΡΠΎΡΠ½ΡΡ
ΠΈΡΠ΅ΡΠ°ΡΠΈΡΡ
Π² ΠΏΠΎΠ΄ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π΅ ΠΡΡΠ»ΠΎΠ²Π°. ΠΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΎ Π²Π»ΠΈΡΠ½ΠΈΠ΅ Π½Π° ΠΈΡΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΡΠΉ ΠΏΡΠΎΡΠ΅ΡΡ Π²ΡΠ΅Ρ
ΡΡΠ΅Ρ
ΡΠΏΠΎΡΠΎΠ±ΠΎΠ² Π΅Π³ΠΎ ΡΡΠΊΠΎΡΠ΅Π½ΠΈΡ ΠΊΠ°ΠΊ ΠΏΠΎ ΠΎΡΠ΄Π΅Π»ΡΠ½ΠΎΡΡΠΈ, ΡΠ°ΠΊ ΠΈ ΠΏΡΠΈ ΠΈΡ
ΠΊΠΎΠΌΠ±ΠΈΠ½ΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ ΠΊΠ°ΠΆΠ΄ΡΠΉ ΠΈΠ· ΡΠΊΠ°Π·Π°Π½Π½ΡΡ
ΡΠΏΠΎΡΠΎΠ±ΠΎΠ² Π²Π½ΠΎΡΠΈΡ ΡΠ²ΠΎΠΉ Π²ΠΊΠ»Π°Π΄ Π² ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π΅Π½Π½ΡΠΉ ΠΏΠΎΠΊΠ°Π·Π°ΡΠ΅Π»Ρ ΡΡΠΊΠΎΡΠ΅Π½ΠΈΡ ΠΈΡΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ ΠΏΡΠΎΡΠ΅ΡΡΠ°. ΠΡΠΈ ΡΡΠΎΠΌ Π½Π°ΠΈΠ±ΠΎΠ»ΡΡΠΈΠΉ Π²ΠΊΠ»Π°Π΄ Π΄Π°Π΅Ρ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ Π°Π»Π³ΠΎΡΠΈΡΠΌΠ°, ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΠ΅Π³ΠΎ ΠΏΠΎΠ΄ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π° ΠΡΡΠ»ΠΎΠ²Π°. ΠΠΎΠΌΠ±ΠΈΠ½ΠΈΡΠΎΠ²Π°Π½Π½ΠΎΠ΅ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ ΠΎΠ΄Π½ΠΎΠ²ΡΠ΅ΠΌΠ΅Π½Π½ΠΎ Π²ΡΠ΅Ρ
ΡΡΠ΅Ρ
ΡΠΏΠΎΡΠΎΠ±ΠΎΠ² ΡΡΠΊΠΎΡΠ΅Π½ΠΈΡ ΠΈΡΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΊΠΎΠ½ΠΊΡΠ΅ΡΠ½ΡΡ
ΠΊΡΠ°Π΅Π²ΡΡ
Π·Π°Π΄Π°Ρ ΠΏΠΎΠ·Π²ΠΎΠ»ΠΈΠ»ΠΎ ΡΠΌΠ΅Π½ΡΡΠΈΡΡ Π²ΡΠ΅ΠΌΡ ΠΈΡ
ΡΠ΅ΡΠ΅Π½ΠΈΡ Π½Π° ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ΅ Π΄ΠΎ 230 ΡΠ°Π· ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΡΠΎ ΡΠ»ΡΡΠ°Π΅ΠΌ, ΠΊΠΎΠ³Π΄Π° Π½ΠΈΠΊΠ°ΠΊΠΈΠ΅ ΡΠΏΠΎΡΠΎΠ±Ρ ΡΡΠΊΠΎΡΠ΅Π½ΠΈΡ Π½Π΅ ΠΏΡΠΈΠΌΠ΅Π½ΡΠ»ΠΈΡΡ. ΠΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ Π΄Π²ΡΡ
ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΏΡΠ΅Π΄ΠΎΠ±ΡΡΠ»Π°Π²Π»ΠΈΠ²Π°ΡΠ΅Π»Ρ. ΠΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½ΠΎ Π½Π°Ρ
ΠΎΠ΄ΠΈΡΡ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΡΠ΅ Π·Π½Π°ΡΠ΅Π½ΠΈΡ Π΅Π³ΠΎ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² ΠΏΡΡΠ΅ΠΌ ΡΠΈΡΠ»Π΅Π½Π½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΎΡΠ½ΠΎΡΠΈΡΠ΅Π»ΡΠ½ΠΎ Π½Π΅ΡΡΡΠ΄ΠΎΠ΅ΠΌΠΊΠΎΠΉ Π·Π°Π΄Π°ΡΠΈ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΡΠΈΡΠ»Π° ΠΎΠ±ΡΡΠ»ΠΎΠ²Π»Π΅Π½Π½ΠΎΡΡΠΈ ΠΌΠΎΠ΄ΠΈΡΠΈΡΠΈΡΠΎΠ²Π°Π½Π½ΠΎΠΉ ΠΏΡΠ΅Π΄ΠΎΠ±ΡΡΠ»Π°Π²Π»ΠΈΠ²Π°ΡΠ΅Π»Π΅ΠΌ ΡΠΈΡΡΠ΅ΠΌΡ Π»ΠΈΠ½Π΅ΠΉΠ½ΡΡ
Π°Π»Π³Π΅Π±ΡΠ°ΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ, ΡΠ΅ΡΠ°Π΅ΠΌΠΎΠΉ Π² ΠΌΠ΅ΡΠΎΠ΄Π΅ ΠΠΠ. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ Π² ΠΌΠ½ΠΎΠ³ΠΎΡΠ΅ΡΠΎΡΠ½ΠΎΠΌ Π²Π°ΡΠΈΠ°Π½ΡΠ΅ ΠΌΠ΅ΡΠΎΠ΄Π° ΠΠΠ Π΄Π»Ρ ΡΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΎΠ³ΠΎ ΡΠΌΠ΅Π½ΡΡΠ΅Π½ΠΈΡ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΊΡΠ°Π΅Π²ΠΎΠΉ Π·Π°Π΄Π°ΡΠΈ Π΄ΠΎΡΡΠ°ΡΠΎΡΠ½ΠΎ ΠΎΠ³ΡΠ°Π½ΠΈΡΠΈΡΡΡΡ ΡΠΎΠ»ΡΠΊΠΎ ΠΏΡΠΎΡΡΠΎΠΉ ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠ΅ΠΉ ΠΏΡΠΎΠ΄ΠΎΠ»ΠΆΠ΅Π½ΠΈΡ ΡΠ΅ΡΠ΅Π½ΠΈΡ Π½Π° ΠΌΠ½ΠΎΠ³ΠΎΡΠ΅ΡΠΎΡΠ½ΠΎΠΌ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ΅. ΠΡΠΈΠ²ΠΎΠ΄ΡΡΡΡ ΠΌΠ½ΠΎΠ³ΠΎΡΠΈΡΠ»Π΅Π½Π½ΡΠ΅ ΠΏΡΠΈΠΌΠ΅ΡΡ ΡΠ°ΡΡΠ΅ΡΠΎΠ², Π΄Π΅ΠΌΠΎΠ½ΡΡΡΠΈΡΡΡΡΠΈΠ΅ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΠΎΡΡΡ ΠΏΡΠ΅Π΄Π»Π°Π³Π°Π΅ΠΌΡΡ
ΠΏΠΎΠ΄Ρ
ΠΎΠ΄ΠΎΠ² ΠΊ ΡΡΠΊΠΎΡΠ΅Π½ΠΈΡ ΠΈΡΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΡΡ
ΠΏΡΠΎΡΠ΅ΡΡΠΎΠ² ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΊΡΠ°Π΅Π²ΡΡ
Π·Π°Π΄Π°Ρ Π΄Π»Ρ Π΄Π²ΡΠΌΠ΅ΡΠ½ΡΡ
ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ ΠΠ°Π²ΡΠ΅βΠ‘ΡΠΎΠΊΡΠ°. Π£ΠΊΠ°Π·ΡΠ²Π°Π΅ΡΡΡ, ΡΡΠΎ ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π½Π°Ρ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΈΡ ΡΠΏΠΎΡΠΎΠ±ΠΎΠ² ΡΡΠΊΠΎΡΠ΅Π½ΠΈΡ ΠΈΡΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΡΡ
ΠΏΡΠΎΡΠ΅ΡΡΠΎΠ² ΠΌΠΎΠΆΠ΅Ρ Π±ΡΡΡ ΡΠ΅Π°Π»ΠΈΠ·ΠΎΠ²Π°Π½Π° ΡΠ°ΠΊΠΆΠ΅ Π² ΡΠ°ΠΌΠΊΠ°Ρ
ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΡ Π΄ΡΡΠ³ΠΈΡ
ΡΠΈΡΠ»Π΅Π½Π½ΡΡ
ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΡΠ΅ΡΠ΅Π½ΠΈΡ PDE