3. The weighted finite-element method

Definition 1

identity

Theorem 1

Comment 1

<sup>2</sup>,αð Þ Ω; δ . In this case

generalized solution [27].

where c3 is a positive constant independent of f.

W�<sup>1</sup>

A function u<sup>v</sup> from the set W<sup>1</sup>

holds for any fixed value of ν satisfying the inequality

298 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

<sup>2</sup>,νð Þ Ω; δ is called an Rv-generalized solution to the problem (2), (3) if it

að Þ¼ uν; v lð Þ v (5)

ν ≥ β: (6)

<sup>2</sup>,νð Þ <sup>Ω</sup> <sup>≤</sup> <sup>c</sup>3k k<sup>f</sup> <sup>L</sup>2,βð Þ <sup>Ω</sup> , (7)

<sup>2</sup>,αð Þ Ω; δ .

<sup>2</sup> (H<sup>1</sup> ), or,

<sup>2</sup>, <sup>ν</sup>ð Þ Ω; δ the integral

satisfies boundary condition (3) almost everywhere on ∂Ω and for every v from W<sup>1</sup>

existence and uniqueness of its Rv-generalized solution were established.

k k u<sup>ν</sup> <sup>W</sup><sup>1</sup>

boundary conditions has a unique Rv-generalized solution u<sup>v</sup> in the set W�<sup>1</sup>

tions) and with smooth enough domain boundary [19–22].

In [17], for the boundary value problem (2)–(3) with homogeneous boundary conditions,

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 u<sup>v</sup> in the set

Then for any ν > β, there always exists parameter δ such that the problem (2)–(3) with homogeneous

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 condi-

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 [23–26]. On the basis of the Galerkin method, theories of difference schemes, finite volumes, and finite-element method were developed to find approximate

Let us call boundary value problem a problem with strong singularity if its generalized

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

solution could not be defined. This solution does not belong to the Sobolev space W<sup>1</sup>

A finite-element scheme for problems (2)–(3) is constructed relying on the definition of an Rνgeneralized solution. For this purpose, a quasi-uniform triangulation T<sup>h</sup> 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 (<sup>k</sup> = 1,…,N), which are triangulation nodes. Denoted by <sup>Ω</sup><sup>h</sup> <sup>¼</sup> <sup>∪</sup><sup>K</sup> <sup>∈</sup>T<sup>h</sup> <sup>K</sup>—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 ¼ f g Pk k¼n k¼1, the set of triangulation internal nodes; by P ¼ f g Pk k¼N <sup>k</sup>¼nþ1, the set of nodes belonging to the <sup>∂</sup>Ω.

Each node Pk ∈P is associated with a function Ψ<sup>k</sup> of the form

$$\Psi\_k(\mathfrak{x}) = \rho^{\nu^\*}(\mathfrak{x})\phi\_k(\mathfrak{x}), \quad k = 1, \dots, n\_\nu$$

where ϕkð Þx is linear on each finite element, ϕ<sup>k</sup> Pj <sup>¼</sup> <sup>δ</sup>kj, k, j <sup>¼</sup> <sup>1</sup>, …, n <sup>δ</sup>kj is the Kronecker delta, and ν<sup>∗</sup> is a real number.

The set <sup>V</sup><sup>h</sup> is defined as the linear span of the system of basis functions f g <sup>Ψ</sup><sup>k</sup> k¼n <sup>k</sup>¼<sup>1</sup> . Denote the corresponding vector set by <sup>V</sup><sup>h</sup> <sup>¼</sup> <sup>V</sup><sup>h</sup> <sup>2</sup> . In set V<sup>h</sup> , one singled out the subset <sup>V</sup>�<sup>h</sup> <sup>¼</sup> <sup>v</sup><sup>∈</sup> <sup>V</sup><sup>h</sup> ; vi ð Þj Pk Pk <sup>∈</sup> <sup>∂</sup><sup>Ω</sup> ¼ 0, i ¼ 1, 2g.

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

$$\mathbf{u}\_{\nu,1}^h = \sum\_{k=1}^n d\_{2k-1} \boldsymbol{\Psi}\_{k\boldsymbol{\nu}} \quad \mathbf{u}\_{\nu,2}^h = \sum\_{k=1}^n d\_{2k} \boldsymbol{\Psi}\_{k\boldsymbol{\nu}} \quad d\_{\boldsymbol{\beta}} = \boldsymbol{\rho}^{-\boldsymbol{\nu}^\*} \left(\boldsymbol{P}\_{\left[\frac{\boldsymbol{\nu}}{\boldsymbol{\pi}}\right]}\right) \mathbf{c}\_{\boldsymbol{\beta}} \quad j = 1, \dots, 2n.$$

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

Each square was subdivided into two triangles by the diagonal.

mesh for N = 4 is presented in Figure 1.

One calculated the errors <sup>e</sup> <sup>¼</sup> ð Þ¼ <sup>e</sup>1;e<sup>2</sup> <sup>u</sup><sup>1</sup> � uh

equality (5) for ν ¼ 0.

<sup>1</sup>, <sup>ν</sup>; uh 2, ν

> 2 k ku <sup>W</sup><sup>1</sup> 2

<sup>2</sup> <sup>η</sup> <sup>¼</sup> k k<sup>e</sup> <sup>W</sup><sup>1</sup>

!

k k e<sup>ν</sup> <sup>W</sup><sup>1</sup> 2,ν k ku <sup>W</sup><sup>1</sup> 2,ν

<sup>u</sup><sup>2</sup> � uh

uh <sup>ν</sup> <sup>¼</sup> <sup>u</sup><sup>h</sup>

W<sup>1</sup>

η<sup>ν</sup> ¼

In this case, size of the mesh-step <sup>h</sup> could be computed by <sup>h</sup> <sup>¼</sup> ffiffiffi

<sup>ν</sup>, <sup>2</sup><sup>Þ</sup> of numerical approximation to the generalized <sup>u</sup><sup>h</sup> <sup>¼</sup> uh

Figure 1. Example of regular mesh (a), and graded meshes I (b) and II (c) (N = 4, κ ¼ 0:4).

"Proba-IV" [31] with regular meshes which were built by the following scheme:

Results of numerical experiments presented in this subsection were obtained using the code

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;

Calculations were performed for different values of N. Optimal parameters δ, ν, and ν<sup>∗</sup> were obtained by the program complex [32]. Generalized solution was determined by the integral

> <sup>1</sup>; <sup>u</sup><sup>2</sup> � <sup>u</sup><sup>h</sup> 2

with different values of h. In addition, these tables contain ratios between error

� � solutions, respectively. Problems A and B in Tables 1 and <sup>4</sup>, respectively,

present values of relative errors of the generalized solution in the norm of the Sobolev space

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 O hð Þ. 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.

� � and the <sup>R</sup>ν-generalized one in the norm of the weighted Sobolev space <sup>W</sup><sup>1</sup>

2

Weighted Finite-Element Method for Elasticity Problems with Singularity

� � and <sup>e</sup><sup>ν</sup> <sup>¼</sup> ð Þ¼ <sup>e</sup>ν, <sup>1</sup>;eν, <sup>2</sup> <sup>u</sup><sup>1</sup> � uh

<sup>1</sup>; uh 2

<sup>p</sup> <sup>=</sup>N. Example of the regular

http://dx.doi.org/10.5772/intechopen.72733

<sup>ν</sup>, <sup>1</sup>;

301

2,ν

�

� � and Rν-generalized

#### Definition 2

An approximate Rν-generalized solution to the problems (2)–(3) by the weighted finite-element method is a function u<sup>h</sup> <sup>ν</sup> ∈ V<sup>h</sup> such that it satisfies the boundary condition (3) in the nodes of the boundary ∂Ω and for arbitrary <sup>v</sup><sup>h</sup>ð Þ<sup>x</sup> <sup>∈</sup> <sup>V</sup><sup>h</sup> and <sup>ν</sup> <sup>&</sup>gt; <sup>β</sup> the integral identity

$$a(\mathbf{u}\_{\mathbf{v}}^h, \mathbf{v}^h) = l(\mathbf{v}^h)\_{\mathbf{v}}$$

holds, where u<sup>h</sup> <sup>ν</sup> <sup>¼</sup> <sup>u</sup><sup>h</sup> <sup>ν</sup>,1; uh ν, 2 � �.

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 O hð Þ when weighted finiteelement 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 ¼ 3 and μ ¼ 5. Two kinds of vector-function u ¼ ð Þ u1; u<sup>2</sup> 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

$$\mu\_1 = \cos\left(\mathbf{x}\_1\right)\cos^2\left(\mathbf{x}\_2\right)\left(\mathbf{x}\_1^2 + \mathbf{x}\_2^2\right)^{0.3051}.$$

$$\mu\_2 = \cos^2\left(\mathbf{x}\_1\right)\cos\left(\mathbf{x}\_2\right)\left(\mathbf{x}\_1^2 + \mathbf{x}\_2^2\right)^{0.3051}.$$

Singularity order of u1, u<sup>2</sup> corresponds to the size of the re-entrant corner γ ¼ 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 W<sup>2</sup> <sup>2</sup>ð Þ Ω

$$\mu\_1 = \cos\left(\mathbf{x}\_1\right)\cos^2\left(\mathbf{x}\_2\right)\left(\mathbf{x}\_1^2 + \mathbf{x}\_2^2\right)^{0.3051} + \left(\mathbf{x}\_1^2 + \mathbf{x}\_2^2\right),$$

$$\mu\_2 = \cos^2\left(\mathbf{x}\_1\right)\cos\left(\mathbf{x}\_2\right)\left(\mathbf{x}\_1^2 + \mathbf{x}\_2^2\right)^{0.3051} + \left(\mathbf{x}\_1^2 + \mathbf{x}\_2^2\right).$$
