The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity

Abstract

Mathematical models of fracture physics and mechanics are boundary value problems for differential equations and systems of equations with a singularity. There are two classes of problems with a singularity: with coordinated and uncoordinated degeneracy of the input data, depending on the behavior of the coefficients of the equation. Finite element methods with the first order of convergence rate $O\left(h\right)$ have been created to find an approximate solution to these problems. We construct a scheme of the weighted finite element method of high degree of accuracy for the boundary value problem with uncoordinated degeneracy of the input data and singularity of the solution. The rate of convergence of an approximate solution of the proposed finite element method to the exact $R_{\nu }$-generalized solution in the weight set $W_{2,\nu +\frac{\beta }{2}+2}^1(\Omega ,\delta )$ is investigated. The estimation of finite element approximation $O\left(h^2\right)$ is established.

1. Introduction

Boundary value problems for differential equations and systems of equations with singularity are mathematical models of physical processes in hydrodynamics, electromagnetism, fracture mechanics and other areas. The singularity of the solution to a boundary value problem can be caused by the internal properties of the solution, by the degeneration of the coefficients and right-hand sides of the equation and the boundary conditions, or the presence of re-entrant corners on the boundary of the domain. A boundary value problem is said to have a weak singularity if its solution belongs to the Sobolev space $W_2^1$, but does not belong to $W_2^2$. We have identified two classes of boundary value problems with coordinated and uncoordinated degeneracy of the input data [1]. Problems with uncoordinated degeneracy of the input data have one order of degeneration of the coefficients of the differential equation. This class includes a large number of problems in physics and fracture mechanics, in which the singularity is caused by the geometry of the domain, the presence of re-entrant corners at the boundary.

The classical finite element method and the difference method cannot provide an acceptable order of convergence rate of an approximate solution to an exact solution of a problem with a singularity [2,3]. The presence of a singularity leads to a loss in the accuracy of the numerical method. Effective numerical methods have been developed for finding a solution with an accuracy of $O\left(h\right)$ (h mesh step) for boundary value problems with a corner singularity; among them are smoothed FEM [4,5,6], meshless/meshfree methods [7,8,9,10], extended FEM (XFEM) [11,12,13] and other methods [14,15,16,17,18,19].

We proposed to define an $R_{\nu }$-generalized solution for boundary value problems with a singularity [20]. If a generalized (weak) solution of the problem exists, then it coincides with the $R_{\nu }$-generalized solution. An important property of an $R_{\nu }$-generalized solution is that it belongs to the spaces $W_{2,\nu }^k$$(k\ge 2)$. This makes it possible to construct numerical methods of a high order of accuracy, i.e., with the rate of convergence of the approximate solution to the exact one $O\left(h^2\right)$. Existence and uniqueness, coercivity and differential properties of the solution in weighted Sobolev spaces were studied in [21,22,23,24,25]. A weighted finite element method for finding an approximate $R_{\nu }$-generalized solution with a convergence rate of $O\left(h\right)$ was developed in [26]. This method has been studied and applied for problems with singularity in elasticity theory [27,28,29], electromagnetism [30,31], and hydrodynamics [32,33,34].

In this paper, we consider the first boundary value problem for a second-order differential equation with uncoordinated degeneracy of the input data and with a strong singularity in a convex two-dimensional domain. A weighted finite element method with a convergence rate $O\left(h^2\right)$ is constructed based on the coercive and differential properties of the $R_{\nu }$-generalized solution. The basis functions of a finite element space contain a singular component. The analysis of the convergence of the approximate solution to the exact solution is carried out in the norm of the weighted Sobolev space.

2. An ${\mathit{R}}_{\mathbf{\nu }}$-Generalized Solution

Let us introduce the following notations:

• $R^2$—a two-dimensional Euclidian space,
• $x=(x_1,x_2)$—an arbitrary point in $R^2$,
• $\Omega$—a convex bounded domain in $R^2$,
• $\partial \Omega$—piecewise smooth boundary of $\Omega$,
• $\overline{\Omega }$—the closure of $\Omega$, i.e., $\overline{\Omega }=\Omega \cup \partial \Omega$,
• $\partial {\Omega }^0$—a set of points $\partial {\Omega }^{\left(i\right)}$$(i=1,\cdots ,n)$ belonging to $\partial \Omega$, including the points of the intersection of its smooth pieces, $\partial {\Omega }^0={\bigcup }_{i=1}^n\partial {\Omega }^{\left(i\right)}$,
• $O_i^{\delta }$—a disk of radius $\delta >0$ with its center in $\partial {\Omega }^{\left(i\right)}$$(i=1,\cdots ,n)$, i.e.,
• $O_i^{\delta }=\{x:\parallel x-\partial {\Omega }^{\left(i\right)}\parallel \le \delta \}$, $O_i^{\delta }\cap O_j^{\delta }\ne \varnothing$, $i\ne j$,
• ${\Omega }_{\delta }={\bigcup }_{i=1}^n(\Omega \cap O_i^{\delta })$,
• $\rho \left(x\right)$—a weight function, which coincides with the distance to $\partial {\Omega }^{\left(i\right)}$ in $O_i^{\delta }$$(i=1,\cdots ,n)$ and equals $\delta$ in $\Omega \setminus {\Omega }_{\delta }$.

Now, we introduce the weighed spaces $H_{2,\alpha }^k(\Omega )$ and $W_{2,\alpha }^k(\Omega )$ with norms

$${\parallel u\parallel }_{H_{2,\alpha }^k(\Omega )}={\left(\sum _{\left|\lambda \right|\le k}\underset{\Omega }{\int }{\rho }^{2(\alpha +|\lambda |-k)}{\left|D^{\lambda }u\right|}^{2}dx\right)}^{1/2},$$

$${\parallel u\parallel }_{W_{2,\alpha }^k(\Omega )}={\left(\sum _{\left|\lambda \right|\le k}\underset{\Omega }{\int }{\rho }^{2\alpha }{\left|D^{\lambda }u\right|}^{2}dx\right)}^{1/2},$$

where $D^{\lambda }=\frac{{\partial }^{\left|\lambda \right|}}{\partial x_1^{{\lambda }_1}\partial x_2^{{\lambda }_2}}$, $\lambda =({\lambda }_1,{\lambda }_2)$, ${\lambda }_i\ge 0$ are integers $(i=1,2)$, $\left|\lambda \right|={\lambda }_1+{\lambda }_2$, $\alpha$ is a non-negative real number, k is a non-negative integer; $dx=d{x}_{1}d{x}_2$. For $k=0$, we will use the notation $H_{2,\alpha }^0(\Omega )=W_{2,\alpha }^0(\Omega )=L_{2,\alpha }(\Omega )$.

We denote by $W_{2,\alpha +l-1}^1(\Omega ,\delta )$$(l\ge 1)$ the set of functions satisfying the following conditions:

(a)

$|D^{k}u|\le C_1{\gamma }^{k}k!{\left({\rho }^{\alpha +k}\right)}^{-1}$ for $x\in {\Omega }_{\delta }$, where $k=0,\cdots ,l$, the constants $C_1$, $\gamma \ge 1$ do not depend on k;

(b)

${\parallel u\parallel }_{L_{2,\alpha }(\Omega \setminus {\Omega }_{\delta })}\ge C_2$, $C_2=const$; with the squared norm

$${\parallel u\parallel }_{W_{2,\alpha +l-1}^1(\Omega ,\delta )}^2=\sum _{\left|\lambda \right|\le l}\parallel {\rho }^{\alpha +l-1}|D^{\lambda }u{|\parallel }_{L_2(\Omega )}^2.$$

(1)

The set ${\mathring{W}}_{2,\alpha }^k(\Omega ,\delta )\subset W_{2,\alpha }^k(\Omega ,\delta )$ is defined as the closure of the set of infinitely differentiable and finite functions in $\Omega$ with norm (1).

Denote by $H_{\infty ,-\alpha }^k(\Omega ,C_3)$ and $W_{\infty ,-\alpha }^k(\Omega ,\delta ,C_4)$ the sets of functions with norms satisfying the inequalities

$${\parallel u\parallel }_{H_{\infty ,-\alpha }^k(\Omega ,C_3)}=\underset{\left|\lambda \right|\le k}{max}\underset{\forall x\in \Omega }{\mathrm{vraimax}}\left|{\rho }^{-\alpha +\left|\lambda \right|}{D}^{\lambda }u\right|\le C_3,$$

$${\parallel u\parallel }_{W_{\infty ,-\alpha }^k(\Omega ,\delta ,C_4)}=\underset{\left|\lambda \right|\le k}{max}\underset{\forall x\in \Omega }{\mathrm{vraimax}}\left|{\rho }^{-\alpha }{D}^{\lambda }u\right|\le C_4,$$

where $C_3$, $C_4$ are positive constants independent of u. For $k=0$, we have $H_{\infty ,-\alpha }^0(\Omega ,C_3)=L_{\infty ,-\alpha }(\Omega ,C_3)$, $W_{\infty ,-\alpha }^0(\Omega ,\delta ,C_4)=L_{\infty ,-\alpha }(\Omega ,\delta ,C_4)$.

Let us consider the first boundary value problem

$$-\sum _{l=1}^2\frac{\partial }{\partial x_l}\left(a_{ll}\frac{\partial u}{\partial x_l}\right)+au=f\mathrm{in}\Omega ,$$

(2)

$$u=0\mathrm{on}\partial \Omega .$$

(3)

Assume that the following conditions are satisfied:

$$a_{ll}\in H_{\infty ,-\beta }^1(\Omega ,C_5),a\in L_{\infty ,-\beta }(\Omega ,C_6),$$

(4)

$$\sum _{l=1}^2{a}_{ll}{\xi }_l^2\ge C_7{\rho }^{\beta }\sum _{l=1}^2{\xi }_l^2,a\ge C_8{\rho }^{\beta }\mathrm{almost}\mathrm{everywhere}\mathrm{on}\Omega ,$$

(5)

$$f\in L_{2,\mu }(\Omega ),$$

(6)

where $C_i$$(i=5,\cdots ,8)$ are positive constants independent of x, ${\xi }_1$, ${\xi }_2$ are any real parameters, $\beta$ is a real number, and $\mu$ is a non-negative real number.

Following [1], we will call the boundary value problem (2), (3) the Dirichlet problem with uncoordinated degeneracy of the input data, if conditions (4)–(6) are met.

Introduce the bilinear and linear forms

$$a_{\Omega }(u_{\nu },v)=\underset{\Omega }{\int }\sum _{l=1}^2\left[a_{ll}{\rho }^{2\nu }\frac{\partial u_{\nu }}{\partial x_l}\frac{\partial v}{\partial x_l}+a_{ll}\frac{\partial {\rho }^{2\nu }}{\partial x_l}\frac{\partial u_{\nu }}{\partial x_l}v+a{\rho }^{2\nu }{u}_{\nu }v\right]dx,$$

$$(f,v)=\underset{\Omega }{\int }{\rho }^{2\nu }fvdx.$$

Definition 1.

A function $u_{\nu }$ in ${\mathring{W}}_{2,\nu +\beta /2}^1(\Omega ,\delta )$ is called an $R_{\nu }$-generalized solution of the Dirichlet problem with uncoordinated degeneracy of the input data if $u_{\nu }=0$ almost everywhere on $\partial \Omega$ and for all v in ${\mathring{W}}_{2,\nu +\beta /2}^1(\Omega ,\delta )$ the identity

$$a_{\Omega }(u_{\nu },v)=(f,v)$$

holds, where ν is arbitrary but fixed and satisfies the inequality

$$\nu \ge \mu +\beta /2.$$

(7)

The following theorems are true.

Theorem 1

([22]). Let conditions (4)–(7) hold and the constant $C_8$ be sufficiently large.

Then there exists a parameter ${\nu }^{*}$ such that the $R_{\nu }$-generalized solution $u_{{\nu }^{*}}$ of the Dirichlet problem with uncoordinated degeneracy of the input data exists and is unique in the set ${\mathring{W}}_{2,{\nu }^{*}+\beta /2}^1(\Omega ,\delta )$ and, moreover,

$$\parallel u_{{\nu }^{*}}{\parallel }_{W_{2,{\nu }^{*}+\beta /2}^1(\Omega ,\delta )}\le C_9{\parallel f\parallel }_{L_{2,\mu }(\Omega ,\delta )},$$

where $C_9$ is a positive constant independent of $u_{{\nu }^{*}}$, f.

Theorem 2

([22]). Let the assumptions of Theorem 1 and the inequality ${\nu }^{*}+\beta /2>2$ are valid.

Then, the $R_{\nu }$-generalized solution $u_{{\nu }^{*}}$ of the Dirichlet problem with uncoordinated degeneracy of the input data belongs to the set $W_{2,{\nu }^{*}+\beta /2+1}^2(\Omega ,\delta )$ and the coercivity inequality

$$\parallel u_{{\nu }^{*}}{\parallel }_{W_{2,{\nu }^{*}+\beta /2+1}^2(\Omega ,\delta )}\le C_{10}{\parallel f\parallel }_{L_{2,\mu }(\Omega ,\delta )}$$

holds with a positive constant $C_{10}$ independent of $u_{{\nu }^{*}}$, f.

Theorem 3.

Let conditions (5) and

$$a_{ll}\in H_{\infty ,-\beta }^{k+1}(\Omega ,C_5^{\prime }),a\in H_{\infty ,-\beta }^k(\Omega ,C_6^{\prime }),f\in W_{2,\mu }^k(\Omega ,\delta ),$$

$${\nu }^{*}\ge \mu +\beta /2,{\nu }^{*}+\beta /2>2,\mu \ge k,k\ge 1$$

be satisfied, and let $C_8$ be a sufficiently large constant.

Then, the $R_{\nu }$-generalized solution of the Dirichlet problem with uncoordinated degeneracy of the input data belongs to the set $W_{2,{\nu }^{*}+\beta /2+k+1}^{k+2}(\Omega ,\delta )$ and the estimate

$$\parallel u_{{\nu }^{*}}{\parallel }_{W_{2,{\nu }^{*}+\beta /2+k+1}^{k+2}(\Omega ,\delta )}\le C_{11}{\parallel f\parallel }_{W_{2,\mu }^2(\Omega ,\delta )}$$

holds, where $C_{11}$ is a positive constant independent of $u_{{\nu }^{*}}$, f.

The proof of the statements of this theorem follows directly from Theorem 3.1 [23]. Further, parameter ${\nu }^{*}$ will be denoted by $\nu$.

3. Auxiliary Statements

Lemma 1

([22]). For any function u in the set $W_{2,\alpha }^1(\Omega ,\delta )$, there exists a parameter ${\alpha }^{*}$ such that

$${\parallel u\parallel }_{L_{2,{\alpha }^{*}-1}({\Omega }_{\delta },\delta )}<C_{12}{\parallel u\parallel }_{L_{2,{\alpha }^{*}}(\Omega ,\delta )},$$

where $0<C_{12}<1$.

Lemma 2.

(A) If $u\in W_{2,{\alpha }^{*}}^1(\Omega ,\delta )$, then ${\rho }^{{\alpha }^{*}}u\in W_{2,0}^1(\Omega ,\delta )$ and there exists a positive constant $C_{13}$ independent of u and such that the estimate

$$\parallel {\rho }^{{\alpha }^{*}}{u\parallel }_{W_{2,0}^1(\Omega ,\delta )}\le C_{13}{\parallel u\parallel }_{W_{2,{\alpha }^{*}}^1(\Omega ,\delta )}$$

is true.

 (B)

If ${\rho }^{{\alpha }^{*}}u\in W_{2,0}^1(\Omega ,\delta )$, then $u\in W_{2,{\alpha }^{*}}^1(\Omega ,\delta )$ and, moreover,

$${\parallel u\parallel }_{W_{2,{\alpha }^{*}}^1(\Omega ,\delta )}\le C_{14}{\parallel {\rho }^{{\alpha }^{*}}u\parallel }_{W_{2,0}^1(\Omega ,\delta )},$$

(8)

where $C_{13}$, $C_{14}$ are positive constants independent of u.

Proof. 

The assertion (A) of Lemma 2 was proved in [22]. Let us prove the statement (B). Using Lemma 1 and $\epsilon$-inequality, we have

$$|{\rho }^{{\alpha }^{*}}{u|}_{W_{2,0}^1(\Omega ,\delta )}\ge C_{15}{\left|u\right|}_{W_{2,{\alpha }^{*}}^1(\Omega ,\delta )}-C_{16}{\parallel u\parallel }_{L_{2,{\alpha }^{*}}(\Omega ,\delta )}.$$

We establish the estimate (8) on the basis of the obtained inequality. □

Lemma 3

([1]). Let k be a non-negative integer:

 (A)

If $u\in H_{2,\alpha }^k(\Omega )$, then ${\rho }^{\alpha -(k-s)}u\in H^s(\Omega )$$(s=0,\cdots ,k)$ and

$$|{\rho }^{\alpha }{u|}_{H^k(\Omega )}+|{\rho }^{\alpha -1}{u|}_{H^{k-1}(\Omega )}+\cdots +\parallel {\rho }^{\alpha -k}{u\parallel }_{L^2(\Omega )}\le C_{17}{\parallel u\parallel }_{H_{2,\alpha }^k(\Omega )},$$

where $C_{17}$ is a positive constant independent of u.

 (B)

If ${\rho }^{\alpha -(k-s)}u\in H^s$$(s=0,\cdots ,k)$, then $u\in H_{2,\alpha }^k(\Omega )$ and there exist positive constants $C_0^{*},\cdots ,C_k^{*}$ independent of u such that

$$C_k^{*}|{\rho }^{\alpha }{u|}_{H^k(\Omega )}+C_{k-1}^{*}|{\rho }^{\alpha -1}{u|}_{H^{k-1}(\Omega )}+\cdots +C_0^{*}\parallel {\rho }^{\alpha -k}{u\parallel }_{L^2(\Omega )}\ge {\parallel u\parallel }_{H_{2,\alpha }^k(\Omega )}.$$

Corollary 1.

If $u\in W_{2,{\alpha }^{*}+s-1}^s(\Omega ,\delta )$$(s=0,\cdots ,k)$, then

$$|{\rho }^{{\alpha }^{*}+k-1}{u|}_{W_{2,0}^k(\Omega ,\delta )}\le C_{18}\left({\left|u\right|}_{W_{2,{\alpha }^{*}+k-1}^k(\Omega ,\delta )}+{\left|u\right|}_{W_{2,{\alpha }^{*}+k-2}^{k-1}(\Omega ,\delta )}+\cdots +{\parallel u\parallel }_{W_{2,{\alpha }^{*}}^1(\Omega ,\delta )}\right),$$

(9)

where $C_{17}$ is a positive constant independent of u.

Proof. 

The statement of Corollary 1 follows from Lemmas 3(A) and 1. □

Remark 1.

Using the technique of proving Lemma 1.3 from [22], it is easy to establish the following estimate

$$\begin{array}{cc}\hfill |{\rho }^{{\alpha }^{*}+k-1}{u|}_{W_{2,0}^k(\Omega ,\delta )}\le & C_k^{\prime }{\left|u\right|}_{W_{2,{\alpha }^{*}+k-1}^k(\Omega ,\delta )}+C_{k-1}^{\prime }{\left|u\right|}_{W_{2,{\alpha }^{*}+k-2}^{k-1}(\Omega ,\delta )}+\cdots +\hfill \\ \hfill +& C_1^{\prime }{\left|u\right|}_{W_{2,{\alpha }^{*}}^1(\Omega ,\delta )}+C_0^{\prime }{\parallel u\parallel }_{W_{2,{\alpha }^{*}}^1(\Omega ,\delta )},\hfill \end{array}$$

where $C_0^{\prime },\cdots ,C_k^{\prime }$ are positive constants independent of u.

Lemma 4.

If $u\in W_{2,{\alpha }^{*}+s-1}^s(\Omega ,\delta )$$(s=1,2,3)$, then ${\rho }^{{\alpha }^{*}+2}{u|}_{\partial {\Omega }^{\left(i\right)}}=0$ and ${\rho }^{{\alpha }^{*}+2}\frac{\partial u}{\partial x_j}{|}_{\partial {\Omega }^{\left(i\right)}}=0$ for $i=1,\cdots ,n$, $j=1,2$.

Proof. 

Since the function u belongs to the set $W_{2,{\alpha }^{*}+s-1}^s(\Omega ,\delta )$$(s=1,2,3)$, then according to Corollary 1 the semi-norm $|{\rho }^{{\alpha }^{*}+s-1}{u|}_{W_{2,{\alpha }^{*}+s-1}^s(\Omega ,\delta )}$ and the norm $\parallel {\rho }^{{\alpha }^{*}+2}{u\parallel }_{L_{2,{\alpha }^{*}+2}(\Omega ,\delta )}$ are finite. From here it is easy to see that ${\rho }^{{\alpha }^{*}+2}u\in W_2^3(\Omega ,\delta )$ and by the Sobolev embedding theorem ${\rho }^{{\alpha }^{*}+2}u\in C^1\left(\overline{\Omega }\right)$.

Let us assume that ${\left({\rho }^{{\alpha }^{*}+2}u\right)}^2{|}_{\partial {\Omega }^{\left(i\right)}}>0$ for some number i$(i=1,\cdots ,n)$. Then, for $\partial {\Omega }^{\left(i\right)}$ there exists such $\epsilon$ that for all x in ${\Omega }_{\partial {\Omega }^{\left(i\right)}}$ the inequality ${\left({\rho }^{{\alpha }^{*}+2}u\right)}^2{|}_{\partial {\Omega }^{\left(i\right)}}>0$ will be true, where ${\Omega }_{\partial {\Omega }^{\left(i\right)}}=\{x\in \Omega ,|x-\partial {\Omega }^{\left(i\right)}|<\epsilon \}$. We have

$$\underset{{\Omega }_{\partial {\Omega }^{\left(i\right)}}}{\int }{\rho }^{2{\alpha }^{*}}{u}^{2}dx=\underset{{\Omega }_{\partial {\Omega }^{\left(i\right)}}}{\int }{\left({\rho }^{{\alpha }^{*}+2}u\right)}^2{\rho }^{-4}dx\ge \underset{{\overline{\Omega }}_{\partial {\Omega }^{\left(i\right)}}}{min}{\left({\rho }^{{\alpha }^{*}+2}u\right)}^2\underset{{\Omega }_{\partial {\Omega }^{\left(i\right)}}}{\int }{\rho }^{-4}dx=\infty .$$

By the condition of Lemma 4, u belongs to $W_{2,{\alpha }^{*}}^1(\Omega ,\delta )$; hence the integral in the left-hand side of the last inequality is finite. A contradiction is obtained; therefore, the assumption that ${\left({\rho }^{{\alpha }^{*}+2}u\right)}^2{|}_{\partial {\Omega }^{\left(i\right)}}>0$ is not true.

By analogy we can prove that ${\rho }^{{\alpha }^{*}+2}\frac{\partial u}{\partial x_j}{|}_{\partial {\Omega }^{\left(i\right)}}=0$ for $i=1,\cdots ,n$, $j=1,2$. □

4. Weighted Finite Element Method

Further, we assume that the coefficients and the right-hand side of the equation of the boundary value problem (2), (3) satisfy the conditions of Theorem 3 for $k=1$. In this case, Theorem 1 and Theorem 2 are valid, the $R_{\nu }$-generalized solution $u_{\nu }$ belongs to the set $W_{2,\nu +\beta /2+2}^3(\Omega ,\delta )$ and estimate

$$\parallel u_{\nu }{\parallel }_{W_{2,\nu +\beta /2+2}^3(\Omega ,\delta )}\le C_{11}{\parallel f\parallel }_{W_{2,\mu }^1(\Omega ,\delta )}$$

(10)

holds.

We construct a finite element method scheme for finding an approximate solution to the boundary value problem (2), (3) from the definition of an $R_{\nu }$-generalized solution.

Let ${\Omega }^{*}$ be an inscribed polygon in $\Omega$, $\partial {\Omega }^{*}$ be the boundary of ${\Omega }^{*}$, and $\partial \mathring{\Omega }$ be a subset of the vertex set of this polygon. Additional rules for constructing the polygon ${\Omega }^{*}$ are specified below in $\left(T_1^h-T_3^h\right)$. Denote by ${\Omega }^{\prime }$ the union of segments formed by pieces of the boundary $\partial \Omega$ and links of the broken line $\partial {\Omega }^{*}$. We perform a quasi-uniform triangulation $T^h$ of the polygon ${\Omega }^{*}$ so that: $\left(T_1^h\right)$ only sides or vertices can be common for the triangles $K_j$$(j=1,\cdots ,N)$, ${\Omega }^h={\bigcup }_{j=1}^N{K}_j$ and ${\Omega }^h\equiv {\Omega }^{*}$, $\partial {\Omega }^{*}$ be the boundary of the domain ${\Omega }^h$, h is the maximal length of the sides of the triangles $K_j$; $\left(T_2^h\right)$$\underset{j=\overline{1,N}}{sup}\frac{h_{max}{K}_j}{h_{min}{K}_j}\le C_{19}$, where $C_{19}$ is a positive constant independent of h; $\left(T_3^h\right)$ all vertices of triangles $K_j$ located on $\partial {\Omega }^h$ belong to $\partial \Omega$; the distance from the points $\partial {\Omega }^h$ to $\partial \Omega$ does not exceed $C_{20}{h}^4$, $C_{20}>0$.

The vertices of triangles (triangulation nodes) will be denoted by $P_1,P_2,\cdots ,P_{N_h}$, where $N_h={\overline{N}}_h+n+\overline{n}$, ${\overline{N}}_h$ is the number of internal triangulation nodes, $\overline{n}$ is the number of nodes belonging to $\partial {\Omega }^h\setminus \partial {\Omega }^0$.

Taking into account the homogeneity of boundary conditions (2) and Lemma 4 we introduce the set basic functions

$${\psi }_i={\rho }^{-(\nu +\beta /2+2)}{\phi }_i,i=\overline{1,3{\overline{N}}_h+2\overline{n}},$$

(11)

where ${\phi }_i$ is a finite function that is a polynomial of the third degree on each triangle whose one of the vertices is the node $P_i$, and the support ${\phi }_i$ consists of these triangles. Functions ${\phi }_i$$\left(i=1,\cdots ,3{\overline{N}}_h+2\overline{n}\right)$ have the following properties:

(1)

for $i=\overline{1,{\overline{N}}_h}$ and $j=\overline{1,{\overline{N}}_h+\overline{n}+n}$

$${\phi }_i\left(P_j\right)=\left\{\begin{array}{cc}1,\hfill & ifi=j,\hfill \\ 0,\hfill & ifi\ne j,\hfill \end{array}\right.\frac{\partial {\phi }_i\left(P_j\right)}{\partial x_1}=\frac{\partial {\phi }_i\left(P_j\right)}{\partial x_2}=0;$$

(2)

for $i=\overline{{\overline{N}}_h+1,2{\overline{N}}_h+\overline{n}}$ and $j=\overline{1,{\overline{N}}_h+\overline{n}+n}$

$$\frac{\partial {\phi }_i\left(P_j\right)}{\partial x_1}=\left\{\begin{array}{cc}1,\hfill & ifi-{\overline{N}}_h=j,\hfill \\ 0,\hfill & ifi-{\overline{N}}_h\ne j,\hfill \end{array}\right.{\phi }_i\left(P_j\right)=\frac{\partial {\phi }_i\left(P_j\right)}{\partial x_2}=0;$$

(3)

for $i=\overline{2{\overline{N}}_h+\overline{n}+1,3{\overline{N}}_h+2\overline{n}}$ and $j=\overline{1,{\overline{N}}_h+\overline{n}+n}$

$$\frac{\partial {\phi }_i\left(P_j\right)}{\partial x_2}=\left\{\begin{array}{cc}1,\hfill & ifi-2{\overline{N}}_h-\overline{n}=j,\hfill \\ 0,\hfill & ifi-2{\overline{N}}_h-\overline{n}\ne j,\hfill \end{array}\right.{\phi }_i\left(P_j\right)=\frac{\partial {\phi }_i\left(P_j\right)}{\partial x_1}=0.$$

We denote by $V^h\left({\Omega }^h\right)$ the linear span ${\left\{{\psi }_i\left(x\right)\right\}}_{i=1}^{3{\overline{N}}_h+2\overline{n}}$. Following [35], we remove the extra degree of freedom in the weighted cubic polynomial $v^h$ from $V^h\left({\Omega }^h\right)$. For this, we require that the coefficients at $x_1^2{x}_2$ and $x_1{x}_2^2$ in the decomposition of $v^h$ on each finite element be equal.

It is obvious that $V^h\left({\Omega }^h\right)\subset W_{\nu +\beta /2+2}^1({\Omega }^h,\delta )$, so any function $v^h$ from $V^h\left({\Omega }^h\right)$ is such that ${\rho }^{\nu +\beta /2+2}{v}^h\in W_{2,0}^1\left({\Omega }^h\right)$ by virtue of Lemma 2(A). Taking into account (11), we note that the function ${\rho }^{\nu +\beta /2+2}{v}^h$ is a cubic spline on ${\Omega }^h$. This function is a polynomial of the third degree on each link l of the broken line $\partial {\Omega }^h$ and vanishes at the ends of l. Let S be a segment formed by a piece of the boundary $\partial \Omega$ and a link l. We extend the function ${\rho }^{\nu +\beta /2+2}{v}^h$ to the segment S while maintaining smoothness (see, for example, Theorem 2 in [36], p. 131) so that the extension belongs to the space $C^1\left(\overline{S}\right)$ and vanishes on the piece of the boundary $\partial \Omega$. In this case, there is an inequality

$$\parallel {\rho }^{\nu +\beta /2+2}{v}^h{\parallel }_{C^1\left(S\right)}\le C_{21}{\parallel {\rho }^{\nu +\beta /2+2}{v}^h\parallel }_{C^1\left(l\right)}.$$

(12)

We build an extension for each segment from $\Omega \setminus {\Omega }^h$. Obviously, the extension ${\rho }^{\nu +\beta /2+2}{v}^h$ belongs to $W_{2,0}^1(\Omega ,\delta )$ and $v^h\in {\mathring{W}}_{\nu +\beta /2+2}^1(\Omega ,\delta )$ by Lemma 2(B). The set of such functions $v^h$ is denoted by $V^h(\Omega ,\delta )$, $V^h(\Omega ,\delta )\subset {\mathring{W}}_{\nu +\beta /2+2}^1(\Omega ,\delta )$.

Definition 2.

A function $u_{\nu }^h$ in the set $V^h(\Omega ,\delta )$ satisfying the equality

$$a_{\Omega }\left(u_{\nu }^h,v^h\right)=\left(f,v^h\right),\forall v^h\in V^h(\Omega ,\delta )$$

is called the approximate $R_{\nu }$-generalized solution of the Dirichlet problem with uncoordinated degeneracy of the input data.

An approximate solution by the finite element method in the polygon ${\Omega }^h$ has the form

$$u_{\nu }^h=\sum _{i=1}^{{\overline{N}}_h}{b}_i{\psi }_i+\sum _{i={\overline{N}}_h+1}^{2{\overline{N}}_h+\overline{n}}{c}_i{\psi }_i+\sum _{i=2{\overline{N}}_h+\overline{n}+1}^{3{\overline{N}}_h+2\overline{n}}{d}_i{\psi }_i,$$

where coefficients

$$\begin{array}{cc}\hfill b_i={\rho }^{\nu +\beta /2+2}\left(P_i\right){\tilde{b}}_i& (i=\overline{1,{\overline{N}}_h}),\hfill \\ \hfill c_i={\rho }^{\nu +\beta /2+2}\left(P_{i-{\overline{N}}_h}\right){\tilde{c}}_i& (i=\overline{{\overline{N}}_h+1,2{\overline{N}}_h+\overline{n}}),\hfill \\ \hfill d_i={\rho }^{\nu +\beta /2+2}\left(P_{i-2{\overline{N}}_h-\overline{n}}\right){\tilde{d}}_i& (i=\overline{2{\overline{N}}_h+\overline{n}+1,3{\overline{N}}_h+2\overline{n}}).\hfill \end{array}$$

The coefficients are defined from the system of equations $a_{\Omega }\left(u_{\nu }^h,{\psi }_i\right)=\left(f,{\psi }_i\right)$

$(i=\overline{1,3{\overline{N}}_h+2\overline{n}})$.

5. The Estimate of the Convergence Rate

We establish an a priori estimate for the error $u_{\nu }-u_{\nu }^h$ in $W_{2,\nu +\beta /2+2}^1(\Omega )$ norm.

Lemma 5.

Let $u_{\nu }$ be the $R_{\nu }$-generalized solution of the boundary value problem (2), (3), and $u_{\nu }^h$ its approximate solution.

Then there exists a positive constant $C_{22}$ independent of the set $V^h(\Omega ,\delta )$ such that the inequality

$$\parallel u_{\nu }-u_{\nu }^h{\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )}\le C_{22}\underset{\forall v^h\in V^h}{inf}{\parallel u_{\nu }-v^h\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )}$$

holds.

Taking into account the continuity on ${\mathring{W}}_{\nu +\beta /2+2}^1(\Omega ,\delta )$ and ${\mathring{W}}_{\nu +\beta /2+2}^1(\Omega ,\delta )$-ellipticity of the bilinear form $a_{\Omega }(·,·)$, the last inequality is established by analogy with [2] (p. 109).

For the function $u_{\nu }$ from the set $W_{2,\nu +\beta /2+2}^3(\Omega ,\delta )$, we construct the interpolant $u_{\nu ,I}$ to domain ${\Omega }^h$

$$\begin{array}{cc}\hfill u_{\nu ,I}=& \sum _{i=1}^{{\overline{N}}_h}{\rho }^{\nu +\beta /2+2}\left(P_i\right)u_{\nu }\left(P_i\right){\psi }_i+\sum _{i={\overline{N}}_h+1}^{2{\overline{N}}_h+\overline{n}}{\rho }^{\nu +\beta /2+2}\left(P_{i-{\overline{N}}_h}\right)\frac{\partial u_{\nu }\left(P_{i-{\overline{N}}_h}\right)}{\partial x_1}{\psi }_i+\hfill \\ \hfill & +\sum _{i=2{\overline{N}}_h+\overline{n}+1}^{3{\overline{N}}_h+2\overline{n}}{\rho }^{\nu +\beta /2+2}\left(P_{i-2{\overline{N}}_h-\overline{n}}\right)\frac{\partial u_{\nu }\left(P_{i-2{\overline{N}}_h-\overline{n}}\right)}{\partial x_2}{\psi }_i\hfill \end{array}$$

and by analogy with Section 4 in the domain ${\Omega }^{\prime }$.

Since

$$\underset{\forall v^h\in V^h}{inf}\parallel u_{\nu }-v^h{\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )}\le {\parallel u_{\nu }-u_{\nu ,I}\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )},$$

we first establish a result about the approximation of $u_{\nu }$ by the interpolant of $u_{\nu ,I}$.

Theorem 4.

Let $u_{\nu }\in W_{2,\nu +\beta /2+2}^3(\Omega ,\delta )$. Then, the estimate

$$\parallel u_{\nu }-u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )}\le C_{23}{h}^2{\parallel f\parallel }_{W_{2,\mu }^1(\Omega ,\delta )}$$

(13)

holds for a given triangulation $T^h$, and the positive constant $C_{23}$ is independent of h, $u_{\nu }$ and f.

Proof. 

We write the equality

$$\parallel u_{\nu }-u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )}=\parallel u_{\nu }-u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{*},\delta )}+{\parallel u_{\nu }-u_{\nu ,I}\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{\prime },\delta )}.$$

(14)

We prove the theorem in two steps. We separately estimate each term in the right-hand side of (14).

Step 1. We establish the estimate $\parallel u_{\nu }-u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{*},\delta )}$.

The functions $u_{\nu }$ and $u_{\nu ,I}$ belong to the set $W_{2,\nu +\beta /2+2}^1({\Omega }^{*},\delta )$.

By Lemma 2(A) ${\rho }^{\nu +\beta /2+2}(u_{\nu }-u_{\nu ,I})\in W_{2,0}^1({\Omega }^{*},\delta )$ and from (8) we have

$$\parallel u_{\nu }-u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{*},\delta )}\le C_{24}{\parallel {\rho }^{\nu +\beta /2+2}\left(u_{\nu }-u_{\nu ,I}\right)\parallel }_{W_{2,0}^1({\Omega }^{*},\delta )}.$$

Taking into account that the domains ${\Omega }^{*}$ and ${\Omega }^h$ coincide, we rewrite the last inequality in the form

$$\parallel u_{\nu }-u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{*},\delta )}\le C_{24}{\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu }-{\rho }^{\nu +\beta /2+2}{u}_{\nu ,I}\parallel }_{W_{2,0}^1({\Omega }^h,\delta )}.$$

(15)

Let K be an arbitrary triangle in ${\Omega }^h$. There are embeddings of the spaces $W_2^3\left(K\right)\subset C^1\left(K\right)$, $W_2^3\left(K\right)\subset W_2^1\left(K\right)$. By Theorem 3.1.6 from [2] (for $p=2$, $q=2$, $m=1$, $k=2$), we have the estimate

$$\parallel {\rho }^{\nu +\beta /2+2}-{\rho }^{\nu +\beta /2+2}{\parallel }_{W_2^1\left(K\right)}\le C_{25}{h}^2{\left|{\rho }^{\nu +\beta /2+2}{u}_{\nu }\right|}_{W_2^3\left(K\right)}$$

(16)

with a positive constant $C_{25}$ independent of ${\rho }^{\nu +\beta /2+2}{u}_{\nu }$ and h.

Summing (16) for all K in ${\Omega }^h$, we obtain

$$\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu }-{\rho }^{\nu +\beta /2+2}{u}_{\nu ,I}{\parallel }_{W_{2,0}^1({\Omega }^h,\delta )}\le C_{26}{h}^2{\left|{\rho }^{\nu +\beta /2+2}{u}_{\nu }\right|}_{W_{2,0}^3({\Omega }^{*},\delta )}.$$

(17)

Let us estimate the semi-norm in the right-hand side of inequality (17). First of all, we note that $u_{\nu }\in W_{\nu +\beta /2+s-1}^s(\Omega ,\delta )$ for $s=1,2,3$. Then, by Corollary 1, we have

$$|{\rho }^{\nu +\beta /2+2}{u}_{\nu }{|}_{W_{2,0}^3({\Omega }^{*},\delta )}\le C_{27}\left(|u_{\nu }{|}_{W_{2,\nu +\beta /2+2}^3({\Omega }^{*},\delta )}+|u_{\nu }{|}_{W_{2,\nu +\beta /2+1}^2({\Omega }^{*},\delta )}+{\parallel u_{\nu }\parallel }_{W_{2,\nu +\beta /2}^1({\Omega }^{*},\delta )}\right).$$

(18)

By using Theorems 1–3 from inequalities (15), (17), and (18), we obtain the estimate

$$\parallel u_{\nu }-u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{*},\delta )}\le C_{28}{h}^2{\parallel f\parallel }_{W_{2,\mu }^1(\Omega ,\delta )}.$$

(19)

Step 2. Let us estimate the second term of the right-hand side of (14).

We can record the inequality

$$\parallel u_{\nu }-u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{\prime },\delta )}\le \parallel u_{\nu }{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{\prime },\delta )}+{\parallel u_{\nu ,I}\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{\prime },\delta )}.$$

(20)

Note that according to $\left(T_3^h\right)$, the area of each segment formed by a piece of the boundary $\partial \Omega$ and a link of the broken line $\partial {\Omega }^{*}$ does not exceed $C_{29}{h}^5$ ($C_{29}$ is a positive constant). Since the number of segments has the order $O\left(h^{-1}\right)$, then $mes{\Omega }^{\prime }=O\left(h^4\right)$.

By virtue of Theorems 1–3, $u_{\nu }\in W_{2,\nu +\beta /2+s-1}^s(\Omega ,\delta )$ for $s=1,2,3$ and $u_{\nu }\in L_{2,\nu +\beta /2+2}(\Omega ,\delta )$. The function ${\rho }^{\nu +\beta /2+2}{u}_{\nu }$ belongs to $W_2^3(\Omega ,\delta )$ by Corollary 1. Since $W_2^3(\Omega ,\delta )\subset C^1(\Omega )$ we have

$$\parallel u_{\nu }{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{\prime },\delta )}\le C_{30}{h}^2\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu }{\parallel }_{C^1(\Omega )}\le C_{31}{h}^2{\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu }\parallel }_{W_2^3(\Omega ,\delta )}.$$

Estimating ${\left|{\rho }^{\nu +\beta /2+2}{u}_{\nu }\right|}_{W_2^s(\Omega ,\delta )}$$(s=1,2,3)$ and $\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu }{\parallel }_{L_{2,\nu +\beta /2+2}(\Omega ,\delta )}$ by analogy with (18), (19) we obtain

$$\parallel u_{\nu }{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{\prime },\delta )}\le C_{31}{h}^2{\parallel f\parallel }_{W_{2,\mu }^1(\Omega ,\delta )}.$$

(21)

Now we estimate $u_{\nu ,I}$ in the norm $W_{2,\mu }^1({\Omega }^{\prime },\delta )$. First, we establish an estimate on a separate segment S, the chord of which is the link l of the broken line $\partial {\Omega }^h$ of length h. Using Lemma 2(B), inequality (12), and taking into account that the area of the segment S is of the order of $O\left(h^5\right)$, we write

$$\parallel u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1(S,\delta )}^2\le C_{32}\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu ,I}{\parallel }_{W_{2,0}^1(S,\delta )}^2\le C_{33}{h}^5{\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu ,I}\parallel }_{C^1\left(l\right)}^2.$$

(22)

Summing (22) over all segments S and taking into account the obvious inequality

$$\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu ,I}{\parallel }_{C^1(\partial {\Omega }^h)}\le C_{34}{\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu }\parallel }_{C^1(\Omega )},$$

we obtain

$$\parallel u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{\prime },\delta )}\le C_{35}{h}^2{\parallel {\rho }^{\nu +\beta /2+2}{u}_{\nu }\parallel }_{C^1(\Omega )}.$$

Taking into account that $W_2^3(\Omega ,\delta )\subset C^1(\Omega )$ and by analogy with (21), we obtain

$$\parallel u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1({\Omega }^{\prime },\delta )}\le C_{36}{h}^2{\parallel f\parallel }_{W_{2,\mu }^1(\Omega ,\delta )}.$$

(23)

From inequalities (19)–(23) and equality (14) the validity of estimate (13) follows. Theorem 4 is proved. □

On the basis of the theorem proved, we establish the estimate of the convergence rate.

Theorem 5.

Suppose that the $R_{\nu }$-generalized solution $u_{\nu }$ of the boundary value problem (2), (3) with uncoordinated degeneracy of the input data belongs to the set $W_{2,\nu +\beta /2+2}^3(\Omega ,\delta )$.

Then there exists a constant $C_{37}$ independent of $u_{\nu }$, $u_{\nu }^h$, f, h such that the convergence estimate

$$\parallel u_{\nu }-u_{\nu }^h{\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )}\le C_{37}{h}^2{\parallel f\parallel }_{W_{2,\mu }^1(\Omega ,\delta )}$$

(24)

holds for the triangulation of the domain Ω constructed.

Proof. 

By using Lemma 5 and Theorem 4, we obtain the estimates

$$\begin{array}{cc}\hfill \parallel u_{\nu }-u_{\nu }^h{\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )}\le & C_{22}\underset{\forall v^h\in V^h}{inf}{\parallel u_{\nu }-v^h\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )}\le \hfill \\ \hfill \le & C_{22}\parallel u_{\nu }-u_{\nu ,I}{\parallel }_{W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )}\le C_{22}{C}_{23}{h}^2{\parallel f\parallel }_{W_{2,\mu }^1(\Omega ,\delta )}.\hfill \end{array}$$

Theorem 5 is proved. □

6. Conclusions

We have constructed a weighted finite element method for the first boundary value problem for a second-order elliptic differential equation with uncoordinated degeneracy of the input data and with a strong singularity. The WFEM scheme is constructed on the basis of the definition of $R_{\nu }$-generalized solution for this problem. Third degree polynomials with weight were chosen as the basis in this method. We have established an estimate for the convergence of an approximate solution to an exact one at a rate of $O\left(h^2\right)$ in the norm of the weighted Sobolev space $W_{2,\nu +\beta /2+2}^1(\Omega ,\delta )$.

In the future, we plan to construct and investigate the convergence of a weighted finite element method of a high order of accuracy for boundary value problems of the Lame system and the biharmonic equation.