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Computer and Information Science » "Finite Element Method - Simulation, Numerical Analysis and Solution Techniques", book edited by Răzvan Păcurar, ISBN 978-953-51-3850-1, Print ISBN 978-953-51-3849-5, Published: February 28, 2018 under CC BY 3.0 license. © The Author(s).

Weighted Finite-Element Method for Elasticity Problems with Singularity

By Viktor Anatolievich Rukavishnikov and Elena Ivanovna Rukavishnikova
DOI: 10.5772/intechopen.72733

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Overview

Figure 1. Example of regular mesh (a), and graded meshes I (b) and II (c) (N = 4, κ = 0.4 ).

Figure 2. Chart of η for the generalized (squared line) and of η ν for R ν -generalized (circled line) (δ=0.0029, ν=1.2, ν*=0.16) solutions to the problem A in dependence on the number of subdivisions 2N.

Figure 3. Chart of η for the generalized (squared line) and of η ν for R ν -generalized (circled line) ( δ = 0.0029 , ν = 1.2 , ν ∗ = 0.16 ) solutions to the problem B in dependence on the number of subdivisions 2N.

Figure 4. Chart of η I for mesh I (squared line) and of η II for mesh II (circled line) for problem A depending on the number of subdivisions 2 N ; κ = 0.3 .

Figure 5. Chart of η I for mesh I (squared line) and of η II for mesh II (circled line) for problem B depending on the number of subdivisions 2 N ; κ = 0.3 .

Weighted Finite-Element Method for Elasticity Problems with Singularity

Viktor Anatolievich Rukavishnikov1, 2 and Elena Ivanovna Rukavishnikova1, 2
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Abstract

In this chapter, the two-dimensional elasticity problem with a singularity caused by the presence of a re-entrant corner on the domain boundary is considered. For this problem, the notion of the Rv-generalized solution is introduced. On the basis of the Rv-generalized solution, a scheme of the weighted finite-element method (FEM) is constructed. The proposed method provides a first-order convergence of the approximate solution to the exact one with respect to the mesh step in the W2,ν1Ω -norm. The convergence rate does not depend on the size of the angle and kind of the boundary conditions imposed on its sides. Comparative analysis of the proposed method with a classical finite-element method and with an FEM with geometric mesh refinement to the singular point is carried out.

Keywords: elasticity problem with singularity, corner singularity, Rv-generalized solution, weighted finite-element method, numerical experiments

1. Introduction

The singularity of the solution to a boundary value problem can be caused by the degeneration of the input data (of the coefficients and right-hand sides of the equation and the boundary conditions), by the geometry of the boundary, or by the internal properties of the solution. The classic numerical methods, such as finite-difference method, finite- and boundary-element methods, have insufficient convergence rate due to singularity which has an influence on the regularity of the solution. It results in significant increase of the computational power and time required for calculation of the solution with the given accuracy. For example, the classic finite-element method allows the finding of the solution for the elasticity problem posed in a two-dimensional domain containing a re-entrant corner of on the boundary with convergence rate O(h1/2). In this case to compute the solution with the accuracy of 10−3 requires a computational power that is one million times greater than in the case of the weighted finite-element method used for the solution of the same problem.

By using meshes refined toward the singularity point, it is possible to construct schemes of the finite-element method with the first order of the rate of convergence of the approximate solution to the exact one [1, 2, 3].

In [4, 5], for boundary value problems with strongly singular solutions for which a generalized solution could not be defined and it does not belong to the Sobolev space H1, it was proposed to define the solution as a Rv-generalized one. The existence and uniqueness of solutions as well as its coercivity and differential properties in the weighted Sobolev spaces and sets were proved [5, 6, 7, 8, 9, 10], the weighted finite-element method was built, and its convergence rate was investigated [11, 12, 13, 14, 15].

In this chapter, for the Lamé system in domains containing re-entrant corners we will state construction and investigation of the weighted FEM for determination of the Rv-generalized solution [16, 17]. Convergence rate of this method did not depend on the corner size and was equal O(h) (see [18], Theorem 2.1). For the elasticity problems with solutions of two types—with both singular and regular components and with singular component only—a comparative numerical analysis of the weighted finite-element method, the classic FEM, and the FEM with meshes geometrically refined toward the singularity point is performed. For the first two methods, the theoretical convergence rate estimations were confirmed. In addition, it was established that FEM with graded meshes failed on high dimensional meshes but weighted FEM stably found approximate solution with theoretical accuracy under the same computational conditions. The mentioned failure can be explained by a small size of steps of the graded mesh in a neighbourhood of the singular point. As a result, for the majority of nodes, the weighted finite-element method allows to find solution with absolute error which is by one or two orders of magnitude less than that for the FEM with graded meshes.

2. Rv-generalized solution

Let Ω=11×1101×10R2 be an L-shaped domain with boundary ∂Ω containing re-entrant corner of 3π/2 with the vertex located in the point O(0,0), Ω¯=Ω∂Ω.

Denote by Ω=xΩ:x12+x221/2δ<1 a part of δ-neighbourhood of the point (0,0) laying in the Ω¯ . A weight function ρ(x) can be introduced that coincides with the distance to the origin in Ω¯ , and equals δ for xΩ¯\Ω¯ .

Let W2,α1Ωδ be the set of functions satisfying the following conditions:

1. Dkuxc1δ/ρxα+k for xΩ¯ , where k = 0,1 and c1 is a positive constant independent on k,

2. uL2,αΩΩc2>0,

with the norm

 uW2,α1Ω=∑∣λ∣≤1∫Ωρ2αDλu2dx1/2, (1)

where Dλ=λ/x1λ1x2λ2 , λ = (λ1,λ2), and |λ|=λ1+λ2; λ1, λ2 are nonnegative integers, and α is a nonnegative real number.

Let L2,αΩδ be the set of functions satisfying conditions (a) and (b) with the norm

uL2,αΩ=Ωρ2αu2dx1/2.

The set W°2,α1ΩδW2,α1Ωδ is defined as the closure in norm (1) of the set C0Ωδ of infinitely differentiable and finite in Ω functions satisfying conditions (a) and (b).

One can say that φW2,α1/2∂Ωδ if there exists a function Φ from W2,α1Ωδ such that Φx∂Ω=φx and

φW2,α1/2∂Ωδ=infΦ∂Ω=ϕΦW2,α1Ωδ.

For the corresponding spaces and sets of vector-functions are used notations W2,α1Ωδ , L2,αΩδ , W°2,α1Ωδ .

Let u = (u1,u2) be a vector-function of displacements. Assume that Ω¯ is a homogeneous isotropic body and the strains are small. Consider a boundary value problem for the displacement field u for the Lamé system with constant coefficients λ and μ:

 −2divμεu+∇λdivu=f,x∈Ω, (2)
 ui=qi,x∈∂Ω, (3)

Here, ε(u) is a strain tensor with components εiju=12uixj+ujxi .

Assume that the right-hand sides of (2), (3) satisfy the conditions

 f∈L2,βΩδ,qi∈W2,β1/2∂Ωδ,i=1,2,β>0. (4)

Denoted by

a1uv=Ωλ+2μu1x1ρ2νv1x1+μu1x2ρ2νv1x2+λu2x2ρ2νv1x1+μu2x1ρ2νv1x2dx,
a2uv=Ωλu1x1ρ2νv2x2+μu1x2ρ2νv2x1+λ+2μu2x2ρ2νv2x2+μu2x1ρ2νv2x1dx,
l1v=Ωρ2νf1v1dx,l2v=Ωρ2νf2v2dx

the bilinear and linear forms and auv=a1uva2uv , lv=l1vl2v .

Definition 1

A function uv from the set W2,ν1Ωδ is called an Rv-generalized solution to the problem (2), (3) if it satisfies boundary condition (3) almost everywhere on ∂Ω and for every v from W2,ν1Ωδ the integral identity

()
 auνv=lv (5)

holds for any fixed value of ν satisfying the inequality

 ν≥β. (6)

In [17], for the boundary value problem (2)–(3) with homogeneous boundary conditions, existence and uniqueness of its Rv-generalized solution were established.

Theorem 1

Let condition (4) be satisfied. Then for any ν>β there always exists parameter δ such that the problem (2)–(3) with homogeneous boundary conditions has a unique Rv-generalized solution uv in the set W°2,α1Ωδ . In this case

()
 uνW2,ν1Ω≤c3fL2,βΩ, (7)

where c3 is a positive constant independent of f.

Then for any ν>β , there always exists parameter δ such that the problem (2)–(3) with homogeneous boundary conditions has a unique Rv-generalized solution uv in the set W°2,α1Ωδ .

Comment 1

At present, there exists a complete theory of classical solutions to boundary value problems with smooth initial data (equation coefficients, right hands of solution and boundary conditions) and with smooth enough domain boundary [19, 20, 21, 22].

()

On the basis of the generalized solution-wide investigations of boundary value problems with discontinuous initial data and not smooth domain boundary were performed in Sobolev and different weighted spaces [23, 24, 25, 26]. On the basis of the Galerkin method, theories of difference schemes, finite volumes, and finite-element method were developed to find approximate generalized solution [27].

Let us call boundary value problem a problem with strong singularity if its generalized solution could not be defined. This solution does not belong to the Sobolev space W21 (H1), or, in other words, the Dirichlet integral of the solution diverges. In [4, 5], we suggested to define a solution to the boundary value problems with strong singularity as an Rv-generalized one in the weighted Sobolev space. The essence of this approach is in introducing weight function into the integral equality. The weight function coincides with the distance to the singular points in their neighbourhoods. The role (sense, mission) of this function is in suppressing of the solution singularity caused by the problem features and is in assuring convergence of integrals in both parts of the integral equality. Taking into account the local character of the singularity, we define weight function as the distance to each singularity point inside the disk of radius δ centered in that points, and outside these disks the weight function equals δ. An exponent of the weight function in the definition of the Rν-generalized solution as well as weighted space containing this solution depend on the spaces to which problem initial data belongs, on geometrical features of the boundary (re-entrant corners), and on changing of the boundary condition type.

In [13, 14], for the transformed system of Maxwell equations in the domain with re-entrant corner in which the solution does not depend on the space W21 , the weighted edge-based finite-element method was developed on the basis of introducing the Rν-generalized solution. Convergence rate of this method is O(h), and it does not depend on the size of singularity as opposed to other methods [28, 29].

The proposed approach of introducing Rν-generalized solution allows to effectively find solutions not only to the boundary value problems with divergent Dirichlet integral but also to problems with weak singularity when the solution belongs to the W21 and does not belong to the space W22 .

3. The weighted finite-element method

A finite-element scheme for problems (2)–(3) is constructed relying on the definition of an Rν-generalized solution. For this purpose, a quasi-uniform triangulation Th of Ω¯ and introduction of special basis functions are constructed.

The domain Ω¯ is divided into a finite number of triangles K (called finite elements) with vertices Pk (k = 1,…,N), which are triangulation nodes. Denoted by Ωh=KThK —the union of all elements; here, h is the longest of their side lengths. It is required that the partition satisfies the conventional constraints imposed on triangulations [10]. Denote by P=Pkk=1k=n , the set of triangulation internal nodes; by P=Pkk=n+1k=N , the set of nodes belonging to the ∂Ω.

Each node PkP is associated with a function Ψk of the form

Ψkx=ρνxϕkx,k=1,,n,

where ϕkx is linear on each finite element, ϕkPj=δkj , k,j=1,,n δkj is the Kronecker delta, and ν is a real number.

The set Vh is defined as the linear span of the system of basis functions Ψkk=1k=n . Denote the corresponding vector set by Vh=Vh2 . In set Vh , one singled out the subset V°h={vVhviPkPk∂Ω=0,i=1,2} .

Associated with the constructed triangulation, the finite-element approximation of the displacement vector components has the form

uν,1h=k=1nd2k1Ψk,uν,2h=k=1nd2kΨk,dj=ρνPj+12cj,j=1,,2n.

Definition 2

An approximate Rν -generalized solution to the problems (2)–(3) by the weighted finite-element method is a function uνhVh such that it satisfies the boundary condition (3) in the nodes of the boundary ∂Ω and for arbitrary vhxVh and ν>β the integral identity

()
auνhvh=lvh,

holds, where uνh=uν,1huν,2h .

In [18], it was shown that convergence rate of the approximate solution to the exact one does not depend on size of the re-entrant corner and is always equal to Oh when weighted finite-element method is used for finding an Rν -generalized solution to elasticity problem. The next section explains results of comparative numerical analysis for the model problems (2)–(3) of the weighted FEM using the classical finite-element method and the FEM with geometrically graded meshes of two kinds.

4. Results of numerical experiments

In the domain, Ω is considered a Dirichlet problem for the Lamé system (2), (3) with constant coefficients λ=3 and μ=5 . Two kinds of vector-function u=u1u2 were used as a solution to the problem.

Problem A

Components of the solution u of the model problem (2), (3) contain only a singular component

()
u1=cosx1cos2x2x12+x220.3051,
u2=cos2x1cosx2x12+x220.3051.

Singularity order of u1, u2 corresponds to the size of the re-entrant corner γ=3π/2 on the domain boundary [30].

Problem B

Solution u of the model problems (2, 3) contains both singular and regular components—regular part belongs to the W22Ω

()
u1=cosx1cos2x2x12+x220.3051+x12+x22,
u2=cos2x1cosx2x12+x220.3051+x12+x22.

4.1. Comparative analysis of the generalized and Rν -generalized solutions

Results of numerical experiments presented in this subsection were obtained using the code ”Proba-IV” [31] with regular meshes which were built by the following scheme:

Domain Ω was divided into squares by lines parallel to coordinate axis, with distance equal to 1/N between them, where N is a half of number of partitioning segments along the greater side;

Each square was subdivided into two triangles by the diagonal.

In this case, size of the mesh-step h could be computed by h=2/N . Example of the regular mesh for N = 4 is presented in Figure 1.

Figure 1.

Example of regular mesh (a), and graded meshes I (b) and II (c) (N = 4, κ=0.4 ).

Calculations were performed for different values of N. Optimal parameters δ, ν , and ν were obtained by the program complex [32]. Generalized solution was determined by the integral equality (5) for ν=0 .

One calculated the errors e=e1e2=u1u1hu2u2h and eν=eν,1eν,2=u1uν,1hu2uν,2h of numerical approximation to the generalized uh=u1hu2h and Rν -generalized uνh=u1,νhu2,νh solutions, respectively. Problems A and B in Tables 1 and 4, respectively, present values of relative errors of the generalized solution in the norm of the Sobolev space W21η=eW21uW21 and the Rν -generalized one in the norm of the weighted Sobolev space W2,ν1 ην=eνW2,ν1uW2,ν1 with different values of h . In addition, these tables contain ratios between error norms, obtained on meshes with step reducing twice. Figures 2 and 3 show the convergence rates of the generalized and Rν -generalized solutions to the corresponding problems with the logarithmic scale. The dashed line in the figures corresponds to convergence with the rate Oh . Tables 2 and 3 (Problem A) and Tables 5 and 6 (Problem B) give limit values: number of nodes where |e1|, |e2|, |ev,1|, and |ev,2| belong to the giving range, this number in percentage to the total number of nodes, and pictures of the absolute error distribution in the domain Ω.

 2N 128 256 512 1024 2048 4096 h 0.01105 0.005524 0.002762 0.001381 0.0006905 0.0003453 η 0.06963 1.52 0.04579 1.52 0.03007 1.52 0.01972 1.53 0.01293 1.53 0.008476 ην 0.07011 1.55 0.04522 1.64 0.02756 2.17 0.01272 2.21 0.005745 1.98 0.002902

Table 1.

Dependence of relative errors of the generalized ( η ) and Rν -generalized ( ην ) ( δ=0.0029 , ν=1.2 , ν=0.16 ) solution to problem A on mesh step.

Figure 2.

Chart of η for the generalized (squared line) and of ην for Rν -generalized (circled line) (δ=0.0029, ν=1.2, ν*=0.16) solutions to the problem A in dependence on the number of subdivisions 2N.

Figure 3.

Chart of η for the generalized (squared line) and of ην for Rν -generalized (circled line) (δ=0.0029 , ν=1.2 , ν=0.16) solutions to the problem B in dependence on the number of subdivisions 2N.

e1 e2 Limit values e1 e2
DistributionNumber%Number
5e6 48.077604557948.0776045579
1e6 29.387369529029.3873695290
5e7 6.7248454686.724845468
1e7 9.62412101929.6241210192
5e8 2.5643224492.564322449
0 3.6244557433.624455743

Table 2.

Number, percentage equivalence, and distribution of nodes where absolute errors ei i=12 of finding components of the approximate generalized solution to problem A are not less than given limit values, 2N=4096 .

e1 e2 Limit values e1 e2
Distribution%Number%Number
5e6 0.03341020.0334102
1e6 0.764960750.76496075
5e7 2.4573089852.457308985
1e7 21.704272918621.7042729186
5e8 12.589158297612.5891582974
0 62.454785339762.4547853399

Table 3.

Number, percentage equivalence, and distribution of nodes where absolute errors eν,i i=12 of finding components of the approximate Rν -generalized solution to problem A (δ=0.0029 , ν=1.2 , ν=0.16) are not less than given limit values, 2N=4096 .

 2N 128 256 512 1024 2048 4096 h 0.01105 0.005524 0.002762 0.001381 0.0006905 0.0003453 η 0.02849 1.54 0.0185 1.53 0.01205 1.53 0.00787 1.53 0.005146 1.53 0.003367 ην 0.02868 1.57 0.01827 1.65 0.01107 2.16 0.005117 2.21 0.002319 1.98 0.001171

Table 4.

Dependence of relative errors of the generalized η and Rν -generalized ην (δ=0.0029 , ν=1.2 , ν=0.16 solution of the problem B on the mesh step.

e1 e2 Limit values e1 e2
DistributionNumber%Number
5e6 48.078604562248.0786045622
1e6 29.387369527829.3873695278
5e7 6.7248454666.724845466
1e7 9.62412101589.6241210159
5e8 2.5643224392.564322438
0 3.6244557583.624455758

Table 5.

Number, percentage equivalence, and distribution of nodes where absolute errors ei i=12 of finding components of the approximate generalized solution to problem B are not less than given limit values, 2N=4096 .

e1 e2 Limit values e1 e2
Distribution%Number%Number
5e6 0.03341080.0334108
1e6 0.771968990.77196899
5e7 2.4813119962.481311996
1e7 21.789273986221.7892739863
5e8 12.588158287612.5881582876
0 62.339783898062.3397838979

Table 6.

Number, percentage equivalence, and distribution of nodes where absolute errors eν,i i=12 ) of finding components of the approximate Rν -generalized solution to problem B ( (δ=0.0029 , ν=1.2 , ν=0.16) are not less than given limit values, 2N=4096 .

4.2. FEM with graded mesh: comparative analysis

This subsection presents results of error analysis for finding generalized solution to the problems A and B by the FEM with graded meshes of two kinds (for detailed information about graded meshes, see [2, 33, 34]).

Mesh I. This partitioning was built by the following scheme

1. In the domain Ω , for a given N, regular mesh was constructed as described in section 4.1.

2. Level l=maxi=1,2Nxi+1N was determined for each node. Here, xi ( i=1,2 ) are initial node coordinates on the regular mesh, means integer part.

3. New coordinates of nodes of the graded mesh are calculated by the formula xi+1NNl1l/N1/κ ( i=1,2 ).

Mesh II. Constructing process for this mesh differs from the one described earlier in the level-calculating mode. Here, l=i=12Nxi+1N . In this case, new coordinates are determined only for nodes with lN .

Examples of meshes I and II are shown in Figure 1(b) and (c), respectively.

The FEM solution obtained on described graded meshes converges with the first rate on the mesh step when the value of the parameter κ is less than the order of singularity [2, 33].

Calculations were performed for different values of N and κ . For each node, one calculated the errors eI=uuIh and eII=uuIIh of the approximate generalized solutions uIh , uIIh obtained on meshes I and II, respectively. The values of relative errors of the generalized solution to the problems A and B in the norm of the Sobolev space W21 for different values of h and κ for mesh I ηI=eIW21uW21 are presented in Tables 7 and 10, respectively, and for mesh II ηII=eIIW21uW21 are presented in Tables 8 and 11, respectively. In addition, these tables contain ratios between error norms and between mesh steps obtained with nodes number increasing four times. Figures 4 and 5 show the convergence rates of the generalized solutions to the corresponding problems for meshes I and II with the logarithmic scale. Dashed line in the figures corresponds to convergence with the rate O(h) as in paragraph 1. Besides, for the problems A and B, Tables 9 and 12, respectively, contain limit values for the following data: number of nodes where e1,II , e2,II belong to the giving range, this number in percentage to the total number of nodes, and pictures of the absolute error distribution in the domain Ω .

 2N 128 256 512 1024 2048 4096 κ=0.3 ηI 2.659e-2 2.00 1.332e-2 2.00 6.675e-3 1.91 3.501e-3 0.75 4.650e-3 0.27 1.741e-2 h 0.062263 1.979 0.031459 1.99 0.015812 1.995 0.007926 1.997 0.003968 1.999 0.001985 κ=0.4 ηI 2.111e-2 2.00 1.057e-2 1.99 5.302e-3 1.78 2.971e-3 0.53 5.559e-3 0.26 2.154e-2 h 0.044928 1.986 0.02262 1.993 0.011349 1.997 0.005684 1.998 0.002845 1.999 0.001423 κ=0.5 ηI 1.990e-2 1.99 1.001e-2 1.99 5.038e-3 1.71 2.940e-3 0.46 6.401e-3 0.25 2.513e-2 h 0.034611 1.99 0.017387 1.995 0.008714 1.998 0.004362 1.999 0.0021823 1.999 0.001092 κ=0.6 ηI 2.315e-2 1.92 1.204e-2 1.93 6.254e-3 1.70 3.678e-3 0.50 7.292e-3 0.26 2.818e-2 h 0.030169 1.993 0.015135 1.997 0.007580 1.998 0.003793 1.999 0.0018973 1.9996 0.0009489

Table 7.

Dependence of relative errors of the generalized solution to problem A with mesh I on the mesh step for different κ .

 2N 128 256 512 1024 2048 4096 κ=0.3 ηII 2.392e-2 2.00 1.196e-2 2.00 5.982e-3 1.99 3.012e-3 1.46 2.059e-3 0.36 5.687e-3 h 0.05114 1.982 0.025805 1.99 0.012962 1.995 0.006496 1.998 0.003252 1.999 0.001627 κ=0.4 ηII 1.974e-2 2.00 9.879e-3 2.00 4.942e-3 1.97 2.511e-3 1.16 2.167e-3 0.30 7.154e-3 h 0.038606 1.988 0.019417 1.994 0.009737 1.997 0.004876 1.999 0.00244 1.999 0.001220 κ=0.5 ηII 1.954e-2 1.98 9.857e-3 1.99 4.963e-3 1.93 2.565e-3 0.94 2.726e-3 0.28 9.725e-3 h 0.031006 1.99 0.015564 1.996 0.007797 1.998 0.003902 1.999 0.001952 1.9995 0.000976 κ=0.6 ηII 2.339e-2 1.91 1.225e-2 1.92 6.386e-3 1.90 3.368e-3 1.14 2.966e-3 0.31 9.712e-3 h 0.025906 1.995 0.012987 1.997 0.006502 1.999 0.003253 1.999 0.001627 1.9997 0.000814

Table 8.

Dependence of relative errors of the generalized solution to problem A with mesh II on the mesh step for different κ .

e1 e2 Limit values e1 e2
Distribution%Number%Number
5e6 0.00160.0016
1e6 35.52427864535.479278292
5e7 13.63110692013.770108011
1e7 33.36326169733.377261808
5e8 7.020550666.98454782
0 10.4618205110.38981486

Table 9.

Number, percentage equivalence, and distribution of nodes where absolute errors ei,II i=12 of finding components of the approximate generalized solution to problem A obtained with mesh II κ=0.5 are not less than given limit values, 2N=1024 .

 2N 128 256 512 1024 2048 4096 κ=0.3 ηI 9.851e-3 1.99 4.955e-3 1.97 2.510e-3 1.36 1.845e-3 0.33 5.639e-3 0.25 2.247e-2 h 0.062263 1.979 0.031459 1.99 0.015812 1.995 0.007926 1.997 0.003968 1.999 0.001985 κ=0.4 ηI 7.712e-3 1.99 3.870e-3 1.95 1.988e-3 0.98 2.034e-3 0.28 7.218e-3 0.25 2.866e-2 h 0.044928 1.986 0.02262 1.993 0.011349 1.997 0.005684 1.998 0.002845 1.999 0.001423 κ=0.5 ηI 7.625e-3 1.99 3.839e-3 1.92 1.995e-3 0.87 2.301e-3 0.27 8.676e-3 0.25 3.454e-2 h 0.034611 1.99 0.017387 1.995 0.008714 1.998 0.004362 1.999 0.002182 1.999 0.001091 κ=0.6 ηI 9.330e-3 1.92 4.849e-3 1.88 2.584e-3 0.91 2.847e-3 0.28 1.016e-2 0.25 4.001e-2 h 0.034611 1.99 0.017387 1.995 0.008714 1.998 0.004362 1.999 0.002182 1.999 0.001091

Table 10.

Dependence of relative errors of the generalized solution to problem B with mesh I on the mesh step for different κ .

 2N 128 256 512 1024 2048 4096 κ=0.3 ηII 5.963e-3 2.00 2.982e-3 2.00 1.492e-3 1.91 7.819e-4 0.77 1.013e-3 0.27 3.757e-3 h 0.05114 1.982 0.025805 1.99 0.012962 1.995 0.006496 1.998 0.003252 1.999 0.0016267 κ=0.4 ηII 6.349e-3 2.00 3.178e-3 2.00 1.591e-3 1.87 8.490e-4 0.67 1.263e-3 0.26 4.805e-3 h 0.038606 1.988 0.019417 1.994 0.009737 1.997 0.004876 1.999 0.00244 1.999 0.0012203 κ=0.5 ηII 7.441e-3 1.98 3.756e-3 1.98 1.894e-3 1.83 1.037e-3 0.60 1.717e-3 0.26 6.606e-3 h 0.031006 1.99 0.015564 1.996 0.007797 1.998 0.003902 1.999 0.001952 1.9995 0.0009763 κ=0.6 ηII 9.574e-3 1.91 5.000e-3 1.92 2.602e-3 1.85 1.409e-3 0.78 1.804e-3 0.27 6.660e-3 h 0.025906 1.995 0.012987 1.997 0.006502 1.999 0.003253 1.999 0.001627 1.9997 0.00081366

Table 11.

Dependence of relative errors of the generalized solution to problem B with mesh II on the mesh step for different κ .

Figure 4.

Chart of ηI for mesh I (squared line) and of ηII for mesh II (circled line) for problem A depending on the number of subdivisions 2N ; κ=0.3 .

Figure 5.

Chart of ηI for mesh I (squared line) and of ηII for mesh II (circled line) for problem B depending on the number of subdivisions 2N ; κ=0.3 .

e1 e2 Limit values e1 e2
Distribution%Number%Number
5e6 0.00160.0016
1e6 23.71818603823.282182623
5e7 18.51814525519.047149398
1e7 34.86427346735.327277097
5e8 8.084634077.89961956
0 14.81611621214.445113305

Table 12.

Number, percentage equivalence, and distribution of nodes where absolute errors ei,II i=12 of finding components of the approximate generalized solution to problem B obtained with mesh II κ=0.5 are not less than given limit values, 2N=1024 .

5. Conclusions

Presented numerical results have demonstrated that:

1. An approximate Rν -generalized solution to the problem (2)–(4) converges to the exact one with the rate Oh in the norm of the set W2,ν1Ωδ in contrast with the generalized solution, which converges with the rate Oh0.61 for the classical FEM;

2. FEM with graded meshes fails on high-dimensional grids because of the small mesh size near the singular point, but the weighted FEM stably allows to find approximate solution with the accuracy Oh under the same computational conditions;

For the approximate Rν -generalized solution obtained by the weighted finite-element method, an absolute error value is by one or two orders of magnitude less than the approximate generalized one obtained by the FEM or by the FEM with graded meshes; this holds for the overwhelming majority of nodes.

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