=Paper=
{{Paper
|id=Vol-2837/short2
|storemode=property
|title=Numerical approach for the one stationary nonlinear problem governing the flow of incompressible viscous fluid in L-shaped domain (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2837/short2.pdf
|volume=Vol-2837
|authors=Alexey V. Rukavishnikov,Viktor A. Rukavishnikov
}}
==Numerical approach for the one stationary nonlinear problem governing the flow of incompressible viscous fluid in L-shaped domain (short paper)==
Numerical approach for the one stationary nonlinear problem governing the flow of incompressible viscous fluid in ๐ฟ-shaped domain Alexey V. Rukavishnikova , Viktor A. Rukavishnikovb a Institute of Applied Mathematics of Far-Eastern Branch, Russian Academy of Sciences, Dzerzhinsky Str., 54, Khabarovsk, 680000, Russia b Computing Center of the Far Eastern Branch of the Russian Academy of Sciences, Kim Yu Chen Str., 65, Khabarovsk, 680000, Russia Abstract The steady Navier-Stokes equations governing the flow of an incompressible viscous fluid in the rotation form in ๐ฟ-shaped domain is considered. The weighted finite element method based on the definition of an ๐ ๐ - generalized solution is constructed. The advantage of the proposed approach over classical approximations is numerically established. The modern elements of computational technologies to find the optimal parameters of the proposed method are used. Keywords corner singularity, weighted finite element method, preconditioning, steady Navier-Stokes equations 1. Introduction Most of the mathematical models representing natural processes are described using boundary value problems for the systems of partial differential equations with a singularity. The peculiarity of the solution is as follows systems in a bounded, connected domain of the Euclidean space ๐2 can be attributed to the presence of obtuse corners on its boundary, to the degeneration of initial data, or to internal characteristics of the solution. If the solution of the boundary value problem does not belong to the Sobolev space ๐21 (ฮฉ), then it is called strong singular. If the solution of the boundary value problem belongs to ๐21 (ฮฉ), but does not belong to ๐22 (ฮฉ), then the boundary value problem is called weakly singular. The generalized solution of such problems in ฮฉ with a boundary containing a reentrant corner ๐ โ (๐, 2๐] belongs to the space ๐21+๐ผโ๐ (ฮฉ), ๐ผ < 1. Moreover, the approximate finite element or finite difference solution by classical method converges to the exact one with a ๎ป(โ๐ผ ) rate. In [1], it was proposed to define the solution of elliptic boundary value problems with a singularity as an ๐ ๐ -generalized one. The approach allows us to introduce a weight space or a set, depending on the geometry of the domain and input data (right-hand sides, equation coefficients, boundary and initial data) to which an ๐ ๐ -generalized solution belongs. In [2, 3, 4], the existence, uniqueness and differential properties of the elliptic problems solution are proved. In [5], a weighted analogue of the Ladyzhenskaya-Babuska-Brezzi condition for the Stokes problem is established. In [6, 7, 8, 9, 10], a weighted finite element method (FEM) for an approximate solution of elliptic problems with a singularity has been developed. Far Eastern Workshop on Computational Technologies and Intelligent Systems, March 2โ3, 2021, Khabarovsk, Russia " 78321a@mail.ru (A.V. Rukavishnikov); vark0102@mail.ru (V.A. Rukavishnikov) 0000-0002-7585-4559 (A.V. Rukavishnikov); 0000-0002-3702-1126 (V.A. Rukavishnikov) ยฉ 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings http://ceur-ws.org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) In the paper, an ๐ ๐ -generalized solution of the steady Navier-Stokes equations governing the flow of an incompressible viscous fluid in the rotation form in ๐ฟ-shaped domain is defined. We use Picardโs iterative procedure [11] to find a solution of a nonlinear problem. Then, we construct a weighted finite element scheme based on the definition of an ๐ ๐ -generalized solution: 1) the functions of the finite element spaces satisfy the mass conservation law in a strong sense โ Scott-Vogelius (SV) pair of 2nd order [12]; 2) basis functions are the product of SV functions and weight functions in some degree. This construction allows us better take into account the behavior of the solution in the ๐ฟ- neighborhood of the singularity point and increase the convergence rate of the approximate solution to the exact one to the first order with respect to the grid step โ, i. e. ๎ป(โ) rate in ๐2,๐1 (ฮฉ) norm. Thus, we develop the numerical method overcomes the so-called pollution effect (see [13]). The same advantage for other hydrodynamics problems was achieved in [14, 15, 16]. The optimal values ๐, ๐ โ and ๐ฟ of the presented weighted FEM using the modern elements of computational technologies were derived numerically. The paper consists of six sections. Section 2 is devoted to the definition of an ๐ ๐ -generalized solu- tion. In Section 3, we present the weighted FEM. The iterative procedure for solving the systems of linear algebraic equations is constructed in Section 4. In Section 5, we shaw and discuss the results of computational experiments. Necessary conclusions are made in last section. 2. The problem statement Let ฮฉ be a bounded, connected domain in the Euclidean space ๐2 . Denote by ฮฉฬ and ฮ the closure and boundary โ of ฮฉ, respectively, ฮฉ = ฮฉ โช ฮ. Let ๐ฑ = (๐ฅ1 , ๐ฅ2 ) be an element of ๐ , where ๐๐ฑ = ๐๐ฅ1 ๐๐ฅ2 and ฬ 2 โ๐ฑโ = ๐ฅ1 + ๐ฅ2 are the measure and norm of ๐ฑ, respectively. 2 2 We write the steady Navier-Stokes equations governing the flow of an incompressible viscous fluid in the convection form: find a velocity field ๐ฎ = ๐ฎ(๐ฑ) = (๐ข1 (๐ฑ), ๐ข2 (๐ฑ)) and a kinematic pressure ๐ = ๐(๐ฑ) from โ ๐ฬ โณ ๐ฎ + (๐ฎ โ โ)๐ฎ + ๐ผ๐ฎ + โ๐ = ๐, div ๐ฎ = ๐ in ฮฉ, (1) ๐ฎ=๐ on ฮ, (2) where ๐ฬ > 0 is the kinematic viscosity coefficient (inversely proportional to the Reynolds number), ๐ผ > 0, ๐ = ๐(๐ฑ) = (๐1 (๐ฑ), ๐2 (๐ฑ)) and ๐ = ๐ (๐ฑ) = (๐1 (๐ฑ), ๐2 (๐ฑ)) are given force field in ฮฉ and boundary data on ฮ, respectively. Denote by โณ, โ and div the Laplace, gradient and divergence operators in ๐2 , respectively. Further, we introduce the necessary notation. Let ๐ฐ = (๐ค1 , ๐ค2 ), ๐ฏ = (๐ฃ1 , ๐ฃ2 ) and ๐ โ scalar, then ๐๐ค1 ๐๐ค2 ๐ฐ โ ๐ฏ = ๐ค1 ๐ฃ1 + ๐ค2 ๐ฃ2 , ๐ ร ๐ฐ = (โ๐๐ค2 , ๐๐ค1 )๐ , curl ๐ฐ = โ + . ๐๐ฅ2 ๐๐ฅ1 We have the identity 1 (๐ฎ โ โ)๐ฎ = ( curl ๐ฎ) ร ๐ฎ + โ๐ฎ2 . (3) 2 It follows from the equality (๐ฐ โ โ)๐ฎ + (๐ฎ โ โ)๐ฐ = โ(๐ฐ โ ๐ฎ) + ( curl ๐ฐ) ร ๐ฎ + ( curl ๐ฎ) ร ๐ฐ and assumption that ๐ฐ = ๐ฎ. Using (3), with ๐ = ๐ + 12 ๐ฎ2 for the system (1), (2) we get the rotation form of the steady Navier- Stokes equations: find a velocity field ๐ฎ and a Bernoulli pressure ๐ such that โ ๐ฬ โณ ๐ฎ + ( curl ๐ฎ) ร ๐ฎ + ๐ผ๐ฎ + โ๐ = ๐, div ๐ฎ = 0 in ฮฉ, (4) ๐ฎ=๐ on ฮ. (5) The system (4), (5) as well as (1), (2) is nonlinear due to the presence of the rotation term ( curl ๐ฎ)ร๐ฎ in the momentum equations. The system on (4), (5) and term in particular we linearized by Picardโs procedure (see [11]). Starting with an initial approximation ๐ฎ(0) for which div ๐ฎ(0) = 0 in ฮฉ and ๐ฎ(0) = ๐ on ฮ (6) Picardโs iteration constructs a sequence of solutions (๐ฎ(๐) , ๐ (๐) ) by solving the linear system: โ ๐ฬ โณ ๐ฎ(๐) + ( curl ๐ฎ(๐โ1) ) ร ๐ฎ(๐) + ๐ผ๐ฎ(๐) + โ๐ (๐) = ๐, div ๐ฎ(๐) = 0 in ฮฉ, (7) ๐ฎ(๐) = ๐ on ฮ. (8) Note that the initial Bernoulli pressure in (7) need not be specified. If ๐ฬ be a not too small and ๐ be a not too large, the steady Navier-Stokes equations (4), (5) have a unique solution (๐ฎ, ๐) and the iterates (๐ฎ(๐) , ๐ (๐) ), ๐ = 1, 2, in (7), (8) converge to it as ๐ โ โ for any choice of the arbitrary ๐ฎ(0) satisfying (6) (see [11]). Note that for a linearized system (7), (8) the conservation laws of the mass and momentum remain in force. In the article, we consider the special case of a bounded polygon domain ฮฉ. Let ฮฉ be a ๐ฟ-shaped domain with one reentrant obtuse corner equals to 3๐2 on the boundary and its vertex coincides with the origin. We define an ๐ ๐ -generalized solution in each Picardโs iteration of the problem (7), (8) and construct the effective weighted FEM. Thus, we solve the nonlinear problem (4), (5) governing the flow of a incompressible viscous fluid in the rotation form and show the advantage of our approximate method over the classical approaches in a ๐ฟ-shaped domain by the computational simulations. Let us introduce the notation and define necessary spaces of generalized functions. Denote by โค ๐ฟ < 1, ๐ฟ > 0} a part of a ๐ฟ-neighborhood of a point (0, 0) contained in ฮฉฬ . Let โฒ ฮฉ๐ฟ = {๐ฑ ฬ { โ ฮฉ โถ โ๐ฑโ โฒ โ๐ฑโ, ๐ฑ โ ฮฉ๐ฟ , ๐(๐ฑ) = โฒ be a weight function. ๐ฟ , ๐ฑ โ ฮฉฬ โงต ฮฉ๐ฟ Denote by ๐ฟ2,๐ฝ (ฮฉ) and ๐2,๐ฝ 1 (ฮฉ) the spaces of functions ๐ฃ(๐ฑ) with a bounded norms โ โ๐ฃโ๐ฟ2,๐ฝ (ฮฉ) = 2๐ฝ 2 โซ ๐ (๐ฑ)๐ฃ (๐ฑ)๐๐ฑ ฮฉ and โ โ๐ฃโ๐2,๐ฝ 1 (ฮฉ) = โ๐ ๐ฝ (๐ฑ)|๐ฃ(๐ฑ)|โ2๐ฟ2 (ฮฉ) + โ๐ ๐ฝ (๐ฑ)|๐ท 1 ๐ฃ(๐ฑ)|โ2๐ฟ2 (ฮฉ) , |๐| respectively, where ๐ท ๐ ๐ฃ(๐ฑ) = ๐๐ฅ ๐๐ 1 ๐๐ฅ๐ฃ ๐2 , |๐| = ๐1 + ๐2 , ๐๐ โฅ 0 - integer. 1 2 Let ๐2,๐ฝ 1 (ฮฉ, ๐ฟ) for ๐ฝ > 0 be a set of functions from the space ๐ 1 (ฮฉ), meets the conditions 2,๐ฝ ๐ฟ ๐ฝ+๐ โฒ โซ ๐ 2๐ฝ (๐ฑ)๐ฃ 2 (๐ฑ)๐๐ฑ โฅ ๐ถ1 > 0, |๐ท ๐ ๐ฃ(๐ฑ)| โค ๐ถ2 ( ๐ฑ โ ฮฉ๐ฟ , (9) ๐(๐ฑ) ) โฒ ฮฉฬ โงตฮฉ๐ฟ where ๐ = 0, 1 and ๐ถ2 a positive constant which is not depend on ๐, with the norm of a space 1 (ฮฉ). Denote by ๐ฟ (ฮฉ, ๐ฟ) a set of functions from the space ๐ฟ (ฮฉ) which subject to conditions (9) ๐2,๐ฝ 2,๐ฝ 2,๐ฝ (only for ๐ = 0) with a norm of a space ๐ฟ2,๐ฝ (ฮฉ). Let ๐ฟ02,๐ฝ (ฮฉ, ๐ฟ) = {๐ โ ๐ฟ2,๐ฝ (ฮฉ, ๐ฟ) โถ โซ ๐ ๐ฝ ๐๐๐ฑ = 0}. ฮฉ ๐ ๐ Let ๐2,๐ฝ 1 (ฮฉ, ๐ฟ) (๐ 1 (ฮฉ, ๐ฟ) โ ๐ 1 (ฮฉ, ๐ฟ)) be a closure by ๐ 1 (ฮฉ) norm of a set of the infinitely- 2,๐ฝ 2,๐ฝ 2,๐ฝ differentiable functions with a compact support in ฮฉ comply with the conditions (9). We will say ๐(๐ฑ) โ ๐2,๐ฝ 1/2 (ฮ, ๐ฟ), if exists a function ฮฆ(๐ฑ) โ ๐2,๐ฝ 1 (ฮฉ, ๐ฟ) such that ฮฆ(๐ฑ)| = ๐(๐ฑ) and โ๐โ ฮ 1/2 ๐2,๐ฝ (ฮ,๐ฟ) = inf โฮฆโ๐2,๐ฝ 1 (ฮฉ) . ฮฆ|ฮ =๐ For the vector field ๐ฏ = (๐ฃ1 , ๐ฃ2 ) we define sets ๐2,๐ฝ (ฮฉ, ๐ฟ) and ๐12,๐ฝ (ฮฉ, ๐ฟ) such that ๐ฃ๐ โ ๐ฟ2,๐ฝ (ฮฉ, ๐ฟ) and โ 1 (ฮฉ, ๐ฟ), respectively, with a bounded norms โ๐ฏโ ๐2,๐ฝ (ฮฉ) = โ๐ฃ1 โ๐ฟ2,๐ฝ (ฮฉ) + โ๐ฃ2 โ๐ฟ2,๐ฝ (ฮฉ) for the first set ๐ฃ๐ โ ๐2,๐ฝ 2 2 โ and โ๐ฏโ๐12,๐ฝ (ฮฉ) = โ๐ฃ1 โ2๐ 1 (ฮฉ) + โ๐ฃ2 โ2๐ 1 (ฮฉ) for the second one. Similarly, we define the sets of vector 2,๐ฝ 2,๐ฝ ๐ fields ๐1/2 ๐ฝ (ฮ, ๐ฟ) and ๐ 1 (ฮฉ, ๐ฟ) on ฮ and in ฮฉ, respectively. 2,๐ฝ We introduce the concept of an ๐ ๐ -generalized solution for the linearized problem (7), (8). ๐ โ ๐2,๐ (ฮฉ, ๐ฟ) and ๐๐ โ ๐ฟ2,๐ (ฮฉ, ๐ฟ) is called an ๐ ๐ -generalized solution Definition 1. The pair ๐ฎ(๐) 1 (๐) 0 ๐ ๐ satisfies a condition (8) on ฮ for any pair ๐ฏ โ๐2,๐ (ฮฉ, ๐ฟ) and ๐ โ of the problem (7), (8), where ๐ฎ(๐) 1 0 ๐ฟ2,๐ (ฮฉ, ๐ฟ) ๐๐ (๐ฎ(๐) (๐) ๐ , ๐ฏ) + ๐(๐ฏ, ๐๐ ) = ๐(๐ฏ), ๐(๐ฎ(๐) ๐ , ๐) = 0 hold, where bilinear and linear forms are as follows ๐ , ๐ฏ) = โซ [๐ผ๐ ๐ฎ๐ โ ๐ฏ + ๐ฬ โ๐ฎ๐ โ โ(๐ ๐ฏ) + ๐ (( curl ๐ฎ๐ ๐๐ (๐ฎ(๐) 2๐ (๐) (๐) 2๐ 2๐ (๐โ1) ) ร ๐ฎ(๐) ๐ ) โ ๐ฏ]๐๐ฑ, ฮฉ ๐(๐ฏ, ๐๐(๐) ) = โ โซ ๐๐(๐) div (๐ 2๐ ๐ฏ)๐๐ฑ, ๐ , ๐) = โ โซ (๐ ๐) div ๐ฎ๐ ๐๐ฑ, ๐(๐ฎ(๐) 2๐ (๐) ๐(๐ฏ) = โซ ๐ 2๐ ๐ โ ๐ฏ๐๐ฑ ฮฉ ฮฉ ฮฉ and ๐ โ ๐2,๐ฝ (ฮฉ, ๐ฟ), ๐ โ ๐1/2 2,๐ฝ (ฮ, ๐ฟ), ๐ โฅ ๐ฝ โฅ 0. 3. The weighted finite element method Perform triangulation ฮฅโ based on the barycentric partition of the elements ๐ฟ๐ of the quasi-uniform triangulation ๐โ of the domain ฮฉ. Then, we divide each element ๐ฟ๐ โ ๐โ (macroelement) into three triangles ๐พ๐๐ (finite element), ๐พ๐๐ โ ฮฅโ (their common vertex is in the barycenter of the macroelement ๐ฟ๐ ). Let ๐ ๐ and ๐๐ be the vertices and midpoints of the sides ๐พ๐ โ ฮฅโ , respectively. Introduce the notation of sets: 1)๐ ๐ฃ๐๐ = ๐ฮฉ๐ฃ๐๐ โช ๐ฮ๐ฃ๐๐ = {๐ ๐ โช ๐๐ }, where ๐ฮฉ๐ฃ๐๐ and ๐ฮ๐ฃ๐๐ are triangulation nodes subsets for the velocity field components in ฮฉ and on ฮ, respectively; 2)๐ ๐๐๐๐ = {๐๐ } of triangulation nodes for the Bernoulli pressure, where the node ๐๐ an exact match to the node ๐ ๐ at the appropriate ๐พ๐๐ . We denote by ฮฉโ = โ ๐พ๐ the totality of the finite elements with sides of order โ. Next, we ๐พ๐ โฮฅโ describe the Scott-Vogelius (SV) element pair (see [12]). For the components of the velocity field, we use polynomials of the second degree (๐ โ ), and for the pressure โ the first one (๐ โ ): ๐ โ = {๐ฃโ โ ๐ถ(ฮฉ) โถ ๐ฃโ |๐พ โ ๐2 (๐พ ), โ๐พ โ ฮฅโ }, ๐โ = ๐ โ ร ๐ โ ; ๐ โ = {๐งโ โ ๐ฟ2 (ฮฉ) โถ ๐งโ |๐พ โ ๐1 (๐พ ), โ๐พ โ ฮฅโ , โซ ๐งโ ๐๐ฑ = 0}. ฮฉ The SV pair has useful feature, namely div ๐โ โ ๐ โ . Next, we represent special basis functions and construct a scheme of the weighted finite element method. To each node ๐๐ โ ๐ฮฉ๐ฃ๐๐ (๐๐ โ ๐ ๐๐๐๐ ) we associate the basis function โ โ ๐๐ (๐ฑ) = ๐ ๐ (๐ฑ) โ ๐๐ (๐ฑ), (๐๐ (๐ฑ) = ๐ ๐ (๐ฑ) โ ๐๐ (๐ฑ)), ๐ = 0, 1, โฆ , ( ๐ = 0, 1, โฆ), { 1, ๐ = ๐, where ๐๐ โ ๐ โ , ๐๐ (๐๐ ) = ๐ฟ๐๐ ๐, ๐ = 0, 1, โฆ (๐๐ โ ๐ โ , ๐๐ (๐๐ ) = ๐ฟ๐๐ , ๐, ๐ = 0, 1, โฆ); ๐ฟ๐๐ = , ๐โ 0, ๐ โ ๐, and ๐ โ are real parameters. The spaces ๐ โ and ๐ โ for the components of the velocity field and pressure are defined as linear span of the basis functions {๐๐ }๐ and {๐๐ }๐ , respectively. Let ๐0โ be a subspace of ๐ โ โถ ๐0โ = {๐ฃ โ โ ๐ โ โถ ๐ฃโ (๐๐ )|๐๐ โ๐ ๐ฃ๐๐ = 0}. The approximate components of the velocity field ๐ฎ(๐) ๐,โ = (๐ข๐,โ,1 , ๐ข๐,โ,2 ) and (๐) (๐) ฮ pressure ๐๐,โ (๐) we seek as a (๐) (๐) (๐) (๐) (๐) ๐ข๐,โ,1 (๐ฑ) = โ ๐2๐ ๐๐ (๐ฑ), ๐ข๐,โ,2 (๐ฑ) = โ ๐2๐+1 ๐๐ (๐ฑ), ๐๐,โ (๐ฑ) = โ ๐๐(๐) ๐๐ (๐ฑ), (10) ๐ ๐ ๐ (๐) ๐ . The coefficients ๐๐ where ๐๐(๐) = ๐ โ๐ (๐[๐/2] ) ๐ฬ ๐ , ๐๐(๐) = ๐ โ๐ (๐๐ ) ๐ฬ (๐) and ๐๐(๐) in (10) are found as โ โ (๐) a result of solving a system (11), (12) (see below). Let ๐โ = ๐ โ ร ๐ โ , ๐โ0 = ๐0โ ร ๐0โ and ๐โ โ ๐ ๐12,๐ (ฮฉโ , ๐ฟ), ๐โ0 โ๐12,๐ (ฮฉโ , ๐ฟ), ๐ โ โ ๐ฟ02,๐ (ฮฉโ , ๐ฟ). Definition 2. The pair ๐ฎ(๐) ๐,โ โ ๐ and ๐๐,โ โ ๐ is called an approximate ๐ ๐ -generalized solution of โ (๐) โ the problem (7), (8) for any pair ๐ฏโ โ ๐โ0 and ๐ โ โ ๐ โ the equalities ๐๐ (๐ฎ(๐) โ โ (๐) โ ๐,โ , ๐ฏ ) + ๐(๐ฏ , ๐๐,โ ) = ๐(๐ฏ ) and ๐(๐ฎ(๐) โ ๐,โ , ๐ ) = 0 (11) hold, where ๐ โ ๐2,๐ฝ (ฮฉ, ๐ฟ), ๐ โ ๐1/2 2,๐ฝ (ฮ, ๐ฟ), ๐ โฅ ๐ฝ โฅ 0. Thus, we construct a weighted FEM to find an ๐ ๐ -generalized solution for the problem (7), (8). We get a system of linear algebraic equation: ๐๐ ๐(๐) + ๐๐(๐) = ๐ and ๐๐ ๐(๐) = ๐, (12) where ๐(๐) = (๐0(๐) , ๐2(๐) , โฆ , ๐1(๐) , ๐3(๐) , โฆ)๐ , ๐(๐) = (๐0(๐) , ๐1(๐) , ๐2(๐) , โฆ)๐ and ๐ be a vector of values of the linear form ๐(๐๐ ). 4. Iterative procedure Now, we present an iterative procedure for solving the sequences of systems view (12), ๐ = 1, 2, 3, โฆ and thus we will approximately solve the original nonlinear problem in rotation form (4), (5): 1. Let ๐(0) and ๐(0) be an arbitrary vectors such that ๐ฎ(0) ๐,โ (๐๐ )|๐๐ โ๐ ๐ฃ๐๐ = ๐ (๐๐ ), div ๐ฎ๐,โ (๐๐ )|๐๐ โ๐ ๐ฃ๐๐ = 0 (0) ฮ (for example ๐ฎ(0)๐,โ (๐๐ )|๐๐ โ๐ฮฉ๐ฃ๐๐ = ๐) and ๐๐,โ (๐๐ )|๐๐ โ๐ (0) ๐๐๐๐ = 0. 2. Realize the Picardโs procedure ๐ = 0, 1, 2, โฆ until the stopping condition is fulfilled: a) Let ๐0(๐) โถ= ๐(๐) and ๐(๐)0 โถ= ๐ ; (๐) b) We construct an internal convergent iterative process (see [17]). For ๐ = 0, 1, โฆ , ๐๐ โ 1 โถ (๐) โ1 ๐๐+1 = ๐๐(๐) + ๐ฬ ๐ (๐ โ ๐๐ ๐๐(๐) โ ๐๐(๐) ๐ ) (๐) ๐๐+1 = ๐(๐) ฬ โ1 ๐ (๐) ๐ + ๐๐ ๐ ๐๐+1 ; c) Let ๐(๐+1) โถ= ๐๐(๐) ๐ and ๐(๐+1) โถ= ๐(๐) ๐๐ , where ๐๐ and ๐๐ are the preconditioning matrices to ๐๐ and ๐๐ = ๐๐ ๐โ1 ฬ ฬ ๐ ๐, respectively. At first, we build a preconditioner ๐ฬ ๐ applying an incomplete LU factorization. We employ the โ1 GMRES(5)-method (see [18]). If we have error ๐ซ0 = ๐ฬ ๐ (๐ฌ โ ๐๐ ๐ฏ) for the problem ๐๐ ๐ฏ = ๐ฌ, then the โ1 โ1 Arnoldi procedure will build an orthonormal basis of the subspace: Span{๐ซ0 , ๐ฬ ๐ ๐๐ ๐ซ0 , โฆ , (๐ฬ ๐ ๐๐ )4 ๐ซ0 }. Further, we construct an auxiliary matrix ๐ฬ ๐ to ๐ฬ ๐ , which is the weight mass matrix ๐๐,๐ ,๐ฬ , such โ that on each ๐ฟ โ ฮฅโ โถ โ 1 โ (๐๐,๐ ,๐ฬ )๐๐ = โซ ๐ 2(๐+๐ ) ๐๐ (๐ฑ) ๐๐ (๐ฑ)๐๐ฑ, ๐, ๐ = 0, 1, โฆ . ๐ฬ ๐ฟ โ โ After that, we define a diagonal matrix ๐ฬ ๐ = ๐ ฬ ๐,๐ ,๐ฬ , where (๐ ฬ ๐,๐ ,๐ฬ ) = โ (๐๐,๐ โ ,๐ฬ ) . ๐๐ ๐๐ ๐ It is known (see [19]), that such diagonal lumping ๐ฬ ๐ is a good preconditioner to matrix ๐ฬ ๐ . In order โ1 to determining the vector ๐ โ โถ= ๐ฬ ๐ ๐ we need to find a solution of the internal procedure: 1) ๐0 = ๐; 2) ๐๐ = ๐๐โ1 + ๐ฬ ๐ (๐ โ ๐ฬ ๐ ๐๐โ1 ) (๐ = 1, โฆ , ๐); โ1 3) ๐ = ๐๐ . โ We use the GMRES(5)-method: (Span{๐ซฬ , (๐ฬ ๐ ๐ฬ ๐ )๐ซฬ , โฆ , (๐ฬ ๐ ๐ฬ )4๐ ๐ซฬ }, ๐ซฬ = ๐ฬ ๐ (๐ โ ๐ฬ ๐ ๐๐โ1 )). โ1 โ1 โ1 5. Results of numerical experiments Let ฮฉ = (โ1, 1)ร(โ1, 1)โงต[0, 1]ร[โ1, 0]. Then we split ฮฉฬ by the horizontal and vertical lines ๐ฅ1(๐) = โ1+๐ โ, and ๐ฅ2(๐) = โ1 + ๐ โ, respectively, into elementary squares ๐๐ , where ๐, ๐ = 0, โฆ , ๐ , โ = ๐2 , ๐ โ even number. After that, we divide each ๐๐ by the diagonal (the lower left corner connects to the upper right corner) into two triangles ๐ฟ๐ (macroelements). Further, each macroelement ๐ฟ๐ is partitioned into three triangles ๐พ๐ (barycentric partition). Consider a solution (๐ฎ, ๐) of nonlinear problem (4), (5) which has a singularity in the vicinity of the reentrant corner ๐ = 32๐ with apex at the origin (0, 0) โถ ๐ ๐ฅ2 ๐ฅ2 ๐ข1 (๐ฅ1 , ๐ฅ2 ) = (๐ฅ12 + ๐ฅ22 ) 2 ((1 + ๐) ๐ธ(๐ฅ1 , ๐ฅ2 ) โ sin(arctg ) + ๐บ(๐ฅ1 , ๐ฅ2 ) โ cos(arctg )), ๐ฅ1 ๐ฅ1 ๐ ๐ฅ2 ๐ฅ2 ๐ข2 (๐ฅ1 , ๐ฅ2 ) = (๐ฅ12 + ๐ฅ22 ) 2 (๐บ(๐ฅ1 , ๐ฅ2 ) โ sin(arctg ) โ (1 + ๐) ๐ธ(๐ฅ1 , ๐ฅ2 ) โ cos(arctg )), ๐ฅ1 ๐ฅ1 ๐โ1 (1 + ๐)2 ๐บ(๐ฅ1 , ๐ฅ2 ) + ๐ป (๐ฅ1 , ๐ฅ2 ) ๐(๐ฅ1 , ๐ฅ2 ) = (๐ฅ12 + ๐ฅ22 ) 2 ( ), ๐โ1 where ๐ฅ2 ๐ฅ2 ๐ธ(๐ฅ1 , ๐ฅ2 ) = cos((1 โ ๐) arctg ) โ cos((1 + ๐) arctg )+ ๐ฅ1 ๐ฅ1 sin((1 + ๐) arctg ๐ฅ๐ฅ21 ) โ cos(๐ ๐) sin((1 โ ๐) arctg ๐ฅ๐ฅ21 ) โ cos(๐ ๐) + + , ๐+1 ๐โ1 ๐ฅ2 ๐ฅ2 ๐บ(๐ฅ1 , ๐ฅ2 ) = (1 + ๐) sin((1 + ๐) arctg ) โ (1 โ ๐) sin((1 โ ๐) arctg )+ ๐ฅ1 ๐ฅ1 ๐ฅ2 ๐ฅ2 +cos((1 + ๐) arctg ) โ cos(๐ ๐) โ cos((1 โ ๐) arctg ) โ cos(๐ ๐), ๐ฅ1 ๐ฅ1 Table 1 The error between the generalized solution ๐ฎโ and exact one ๐ฎ in the ๐12 (ฮฉ) norm. N= 280 140 70 1.379e-1 1.988e-1 2.848e-1 Table 2 The error between an ๐ ๐ -generalized solution ๐ฎ๐,โ and exact one ๐ฎ in the ๐12,๐ (ฮฉ) norm, for different values ๐, ๐ฟ and ๐ โ (๐ โ = ๐ โ ). (๐, ๐ โ , ๐ฟ), ๐ = 280 140 70 (2.0, โ0.5, 0.015) 1.627e-5 3.295e-5 6.629e-5 (2.0, ๐ โ 1, 0.015) 1.394e-5 2.796e-5 5.626e-5 (2.0, โ0.4, 0.015) 1.181e-5 2.355e-5 4.760e-5 (1.9, โ0.5, 0.016) 2.785e-5 5.624e-5 1.131e-4 (1.9, ๐ โ 1, 0.016) 2.331e-5 4.672e-5 9.368e-5 (1.9, โ0.4, 0.016) 1.869e-5 3.759e-5 7.549e-5 Table 3 The number of grid nodes ๐๐ โ ๐ฮฉ๐ฃ๐๐ (in percentage of their total number), where the errors ๐ฝ๐๐ are more than the given limit values ๐๐ , ๐ = 1, 2, of the generalized solution ๐ = ๐ โ = 0, ๐ฟ = 1. N= 280 140 ๐1 = 10โ5 64.51% 77.82% ๐2 = 10โ6 90.97% 94.81% Table 4 The number of grid nodes ๐๐ โ ๐ฮฉ๐ฃ๐๐ (in percentage of their total number), where the errors ๐ฝ๐,๐ ๐ are more than โ the given limit values ๐๐ , ๐ = 1, 2, of an ๐ ๐ - generalized solution ๐ = 1.8, ๐ = โ0.31, ๐ฟ = 0.014. N= 280 140 ๐1 = 10โ5 28.33% 54.30% ๐2 = 10โ6 77.32% 85.76% ๐ฅ2 ๐ฅ2 ๐ป (๐ฅ1 , ๐ฅ2 ) = (1 โ ๐)3 sin((1 โ ๐) arctg ) โ (1 + ๐)3 sin((1 + ๐) arctg )+ ๐ฅ1 ๐ฅ1 ๐ฅ ๐ฅ +(๐ โ 1)2 cos((1 โ ๐) arctg ) โ cos(๐ ๐) โ (๐ + 1)2 cos((1 + ๐) arctg ) โ cos(๐ ๐). 2 2 ๐ฅ1 ๐ฅ1 Let ๐ผ = ๐ฬ = 1 and ๐ = 0.54448. The pair of functions (๐ฎ, ๐) is analytic in ฮฉฬ โงต (0, 0), but ๐ฎ โ ๐22 (ฮฉ) and ๐ โ ๐21 (ฮฉ). It is a typical situation in non-convex polygonal domains. Numerical experiments were carried out on grids with different steps โ. The errors of the gen- eralized (classical FEM with ๐ = 0, ๐ฟ = 1, ๐ โ = ๐ โ = 0) and ๐ ๐ -generalized (presented weighted FEM) solutions were determined using the modulus of the difference between the exact solution and approximate one at the nodes ๐๐ , i. e. ๐ฝ๐๐ = |๐ข๐ (๐๐ ) โ ๐ขโ,๐ (๐๐ )| for the generalized solution and ๐,โ,๐ (๐๐ )| for an ๐ ๐ -generalized one, where ๐๐ โ ๐ฮฉ , ๐ = 1, 2, and also in the norms ๐ = |๐ข (๐ ) โ ๐ข ๐ฝ๐,๐ ๐ฃ๐๐ ๐ ๐ of generalized functions. See Figures 1-2 and Tables 1-4. The optimal values of parameters ๐, ๐ โ and ๐ฟ were derived numerically. Figure 1: Distribution of the points ๐๐ with errors ๐ฝ of the generalized solution (๐ = 0, ๐ฟ = 1, ๐ โ = ๐ โ = 0): ๐) ๐ = 140, ๐) ๐ = 280 and ๐) ๐ = 140, ๐) ๐ = 280 for the 1st and 2nd components of ๐ฎโ , respectively. 6. Conclusions The results of computational experiments for the steady Navier-Stokes equations (4), (5) lead to the following conclusions: 1) An approximate ๐ ๐ -generalized solution by the weighted FEM converges to the exact one with a ๎ป(โ) rate in the ๐12,๐ (ฮฉ) norm (see Table 2), while the approximate generalized solution by the classical FEM converges to the exact one with a ๎ป(โ0.54 ) rate in the ๐12 (ฮฉ) norm (see Table 1). In other words, the proposed method suppresses the so-called pollution effect [13]. 2) For all values of ๐ฟ, ๐ and ๐ โ from the range of optimal values (๐ฟ โผ โ, ๐ โผ 2 and ๐ โ โผ 1 โ ๐) an approximate ๐ ๐ -generalized solution converges to the exact one with a ๎ป(โ) rate in the ๐12,๐ (ฮฉ) norm. 3) The number of nodes and their surroundings by using a weighted FEM, in which the values of the absolute errors ๐ฝ๐,๐ ๐ , ๐ = 1, 2, do not exceed the given values, increases with ๐ and is much more then by using the classical FEM (see Tables 3-4) and Figures 1-2. Acknowledgments The reported study was supported by RSF according to the research project No. 21-11-00039. Compu- tational resources for the numerical experiments were provided by the Shared Services Center "Data Center of FEB RAS". Figure 2: Distribution of the points ๐๐ with errors ๐ฝ๐ of the ๐ ๐ -generalized solution (๐ = 1.9, ๐ฟ = 0.014, ๐ โ = ๐ โ = โ0.35): ๐) ๐ = 140, ๐) ๐ = 280 and ๐) ๐ = 140, ๐) ๐ = 280 for the 1st and 2nd components of ๐ฎ๐,โ , respectively. References [1] V. A. Rukavishnikov, On the differential properties of ๐ ๐ -generalized solution of Dirichlet prob- lem, Dokl. Akad. Nauk SSSR 309 (1989) 1318โ1320. [2] V. A. Rukavishnikov, On the existence and uniqueness of an ๐ ๐ -generalized solution of a bound- ary value problem with uncoordinated degeneration of the input data, Doklady Mathematics 90 (2014) 562โ564. doi:10.1134/S1064562414060155. [3] V. A. Rukavishnikov, E. V. Kuznetsova, The ๐ ๐ -generalized solution of a boundary value problem with a singularity belongs to the space ๐2,๐+๐ฝ/2+๐+1 ๐+2 (ฮฉ, ๐ฟ), Differential Equations 45 (2009) 913โ 917. doi:10.1134/S0012266109060147. [4] V. A. Rukavishnikov, S. G. Nikolaev, On the ๐ ๐ -generalized solution of the Lamรฉ system with cor- ner singularity, Doklady Mathematics 92 (2015) 421โ423. doi:10.1134/S1064562415040080. [5] V. A. Rukavishnikov, A. V. Rukavishnikov, Weighted finite element method for the Stokes prob- lem with corner singularity, Journal of Computational and Applied Mathematics 341 (2018) 144โ156. doi:10.1016/j.cam.2018.04.014. [6] V. A. Rukavishnikov, A. Y. Bespalov, An exponential rate of convergence of the finite element method for the Dirichlet problem with a singularity of the solution, Doklady Mathematics 62 (2000) 266โ270. [7] V. A. Rukavishnikov, E. V. Kuznetsova, A finite element method scheme for boundary value problems with noncoordinated degeneration of input data, Numerical Analysis and Applications 2 (2009) 250โ259. doi:10.1134/S1995423909030069. [8] V. A. Rukavishnikov, E. I. Rukavishnikova, Numerical method for Dirichlet problem with de- generation of the solution on the entire boundary, Symmetry 11 (2019) 1455. doi:10.3390/ sym11121455. [9] V. A. Rukavishnikov, A. O. Mosolapov, E. I. Rukavishnikova, Weighted finite element method for elasticity problem with a crack, Computers and Structures 243 (2021) 106400. doi:10.1016/ j.compstruc.2020.106400. [10] V. A. Rukavishnikov, O. P. Tkachenko, Dynamics of a fluid-filled curvilinear pipeline, Applied Mathematics and Mechanics 39 (2018) 905โ922. doi:10.1007/s10483-018-2338-9. [11] M. Benzi, H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numerica 14 (2005) 1โ137. doi:10.1017/S0962492904000212. [12] L. R. Scott, M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, Mathematical Modelling and Numerical Analysis 19 (1985) 111โ143. [13] H. Blum, The influence of reentrant corners in the numerical approximation of viscous flow problems, volume 30 of Numerical Treatment of the Navier-Stokes Equations, Springer, 1990. doi:110.1007/978-3-663-14004-7-4. [14] V. A. Rukavishnikov, A. V. Rukavishnikov, New approximate method for solving the Stokes problem in a domain with corner singularity, Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming & Computer Software 11 (2018) 95โ108. doi:10.14529/ mmp180109. [15] V. A. Rukavishnikov, A. V. Rukavishnikov, New numerical method for the rotation form of the Oseen problem with corner singularity, Symmetry 11 (2019) 54. doi:10.3390/sym11010054. [16] V. A. Rukavishnikov, A. V. Rukavishnikov, The method of numerical solution of the one sta- tionary hydrodymics problem in convective form in ๐ฟ-shaped domain, Computer Research and Modeling 12 (2020) 1291โ1306. doi:10.20537/2076-7633-2020-12-6-1291-1306. [17] J. H. Bramble, J. E. Pasciak, A. T. Vassilev, Analysis of the inexact Uzawa algorithm for sad- dle point problems, SIAM Journal on Numerical Analysis 34 (1997) 1072โ1092. doi:10.1137/ S0036142994273343. [18] Y. Saad, Iterative methods for sparse linear systems, 2nd. ed., SIAM, Philadelphia, 2003. [19] M. Olshanskii, A. Reusken, Analysis of a Stokes interface problem, Numerische Mathematik 103 (2006) 129โ149. doi:10.1007/S00211-005-0646-X.