**1. Introduction**

As is known, classical solutions for boundary value problems for elliptic equations with discontinuous coefficients do not exist. Therefore, the notion of a generalized (weak) solution was introduced. Based on this definition and on the Galerkin method, numerous numerical methods were developed for finding approximate solutions of such problems. However, these methods for boundary value problems with singularity lose accuracy, which depends on the smoothness of the solution of the original differential problem (see, for example, [1,2]). The singularity of the solution of the boundary value problem can be caused by the presence of re-entrant corners on the domain boundary, by the degeneration of the coefficients and right-hand sides of equation and boundary conditions, or by the internal properties of the solution (see, for example, [3–9]). For boundary problems with a singularity, we proposed to determine an *Rν*-generalized solution. The existence, uniqueness and differential properties of this kind of solution in the weighted Sobolev spaces were studied in [10–15]. Based the *Rν*-generalized solution, a weighted finite element method was developed for boundary value problems for elliptic equations in two-dimensional domain with a singularity in a finite set of boundary points [16–20]. A weighted FEM was constructed and studied for the Lamé system in a domain with re-entrant corners [21,22]. To find an approximate solution of Maxwell's equations in an L-shaped domain, a weighted edge-based finite element method was proposed in [23,24]. In [25,26] a weight analogue of the condition of Ladyzhenskaya-Babuška-Brezzi was proved, a numerical method was developed for the Stokes and Oseen problems in domains with corner singularity. The main feature of all the developed methods is the convergence of the approximate solution to the exact one with the rate *O*(*h*) in the norms of the Sobolev and Monk weighted spaces, regardless of the reasons causing the solution singularity and its value.

In this paper we consider the Dirichlet problem for an elliptic equation with degeneration of the solution on the entire boundary of a two-dimensional domain. In [27] a finite element method was constructed for this problem and the convergence of this method was established. The paper [28] singles out the weighted subspace of functions for which the approximate solutions converge to an exact solution with a speed *O*(*h*) on a mesh with a special compression of nodes close to the boundary (see [29]). The compression parameters depend on the constructed subspace. Our method of constructing mesh with a special compression of nodes differs from the methods proposed by other authors (see, for example, [30–32]).

Here we test model problems with singularities in a symmetric domain. We carry out a comparative numerical analysis of finite element methods on quasi-uniform meshes and meshes with a special compression of nodes close to the boundary. We obtain experimental confirmation of theoretical estimates and demonstrate the advantage of the proposed method over the classical finite element method. By analogy with [22], we found that it is impossible to use FEM with a strong thickening of mesh, and introduction of an R-generalized solution is required. The existence and uniqueness of the R-generalized solution for this problem were proved in [33].

### **2. Problem Formulation**

Let Ω be a bounded convex two-dimensional domain with twice differentiable boundary *∂*Ω, and let Ω be the closure of Ω, i.e., Ω = Ω ∪ *∂*Ω; *x* = (*x*1, *x*2) and *dx* = *dx*1*dx*2.

We assume that a positive function *ρ*(*x*) belongs to the space *C*(2)(Ω) and coincides in the boundary strip of width *d* > 0 with the distance from *x* (*x* ∈ Ω) to the boundary *∂*Ω.

We introduce the weighted Sobolev space *W<sup>s</sup>* 2,*η*(Ω) with the norm

$$||\upsilon||\_{W^{\mathfrak{s}}\_{2,\eta}(\Omega)} = ||\upsilon||\_{L\_2(\Omega)} + \sum\_{\substack{\mathfrak{m}\_1,\mathfrak{m}\_2 = 0\\|\mathfrak{m}| = s}} \left\| \rho^{-\eta} \frac{\partial^{|\mathfrak{m}|\_{\mathcal{D}}}}{\partial \mathfrak{x}\_1^{\mathfrak{m}\_1} \partial \mathfrak{x}\_2^{\mathfrak{m}\_2}} \right\|\_{L\_2(\Omega)}.$$

where *η* is a real number satisfying the inequalities <sup>1</sup> <sup>2</sup> <sup>−</sup> *<sup>s</sup>* <sup>&</sup>lt; *<sup>η</sup>* <sup>&</sup>lt; <sup>1</sup> <sup>2</sup> ; *s* = 1, 2; *m* = (*m*1, *m*2), |*m*| = *m*<sup>1</sup> + *m*2, *m*1, *m*<sup>2</sup> are integer nonnegative numbers.

Let

$$\mathring{\mathcal{W}}\_{2,\eta}^{\mathfrak{s}}(\Omega) = \{ v \colon v \in \mathcal{W}\_{2,\eta}^{\mathfrak{s}}(\Omega), \quad v|\_{\partial\Omega} = 0 \}.$$

We denote by *L*2,−1−*η*(Ω) the space of functions *f* with the norm

$$\|f\|\_{L\_{2,-1-\eta}(\Omega)} = \left(\int\_{\Omega} |\rho^{1+\eta}f|^2 \,d\mathfrak{x}\right)^{1/2}.$$

We consider the first boundary value problem for a second order elliptic equation

$$-\sum\_{k=1}^{2} \frac{\partial}{\partial \mathbf{x}\_k} \left( a\_{kk}(\mathbf{x}) \frac{\partial \boldsymbol{u}}{\partial \mathbf{x}\_k} \right) + a(\mathbf{x}) \boldsymbol{u} = \boldsymbol{f} \quad \text{in } \Omega,\tag{1}$$

$$\boldsymbol{u} = \boldsymbol{0} \quad \text{on } \partial \Omega.$$

We suppose that the input data of Equation (1) satisfy the conditions:

(a)

$$f \in L\_{2, -1-a}(\Omega),$$

(b) *akk*(*x*) (*k* = 1, 2) are differentiable functions on Ω, such that the inequalities

$$|a\_{kk}(\mathbf{x})| \le C\_1 \rho^{-2a}(\mathbf{x}),\tag{3}$$

$$\left|\left|\frac{\partial a\_{kk}(\mathbf{x})}{\partial \mathbf{x}\_{1}}\right|, \left|\frac{\partial a\_{kk}(\mathbf{x})}{\partial \mathbf{x}\_{2}}\right|\right| \leq C\_{2} \rho^{-2a-1}(\mathbf{x}),\tag{4}$$

$$\sum\_{k=1}^{2} a\_{kk}(\mathbf{x}) \mathfrak{J}\_k^2 \ge C\_3 \rho^{-2a}(\mathbf{x}) \sum\_{k=1}^{2} \mathfrak{J}\_{k\prime}^2 \quad \mathbf{x} \in \Omega, \quad \mathbb{C}\_3 > 0,\tag{5}$$

hold,

(c) the function *a*(*x*) satisfies the inequalities

$$0 < a(\mathbf{x}) \le \mathbb{C}\_4 \rho^{-2a-2}(\mathbf{x}), \quad \mathbf{x} \in \Omega. \tag{6}$$

Here *Ci*, (*i* = 1, ... , 4) are constants independent of *x*, *ξ*<sup>1</sup> and *ξ*<sup>2</sup> are any real parameters, *α* ∈ - <sup>−</sup> <sup>1</sup> 2 , 1 2 .

**Remark 1.** *If Conditions (2)–(6) are fulfilled for the input data, Equation (1) is called a Dirichlet boundary value problem for an elliptic equation with degeneration of the solution on the entire boundary of a two-dimensional domain. Such problems are encountered in gas dynamics, electromagnetism and other subject areas of mathematical physics. The differential properties of solutions of problems with degeneracy on the entire boundary were studied, for the first time, in [7–9].*

We introduce the bilinear and linear forms

$$E(u, w) = \int\_{\Omega} \left( \sum\_{k=1}^{2} a\_{kk}(\mathbf{x}) \frac{\partial u}{\partial \mathbf{x}\_k} \frac{\partial w}{\partial \mathbf{x}\_k} + a(\mathbf{x}) uw \right) d\mathbf{x}, \qquad (f, w) = \int\_{\Omega} fw \, d\mathbf{x}.$$

A function *u* in *W*˚ <sup>1</sup> 2,*α*(Ω) is called a generalized solution of the first boundary value Equation (1) if for any *w* in *W*˚ <sup>1</sup> 2,*α*(Ω) the identity

$$E(u, w) = (f, w)$$

holds.

We note that if Conditions (2)–(6) are satisfied, then there exists a unique generalized solution of the Equation (1) in the space *W*˚ <sup>1</sup> 2,*α*(Ω) (see Theorem 1 from [8]). In addition *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*˚ <sup>2</sup> 2,*α*−1(Ω) (see Theorem 1 from [9]). Moreover, if the function *f* ∈ *L*2,−1−*α*+*λ*(Ω) ⊂ *L*2,−1−*α*(Ω) - <sup>−</sup> <sup>1</sup> <sup>2</sup> <sup>&</sup>lt; *<sup>α</sup>* <sup>&</sup>lt; *<sup>α</sup>* <sup>+</sup> *<sup>λ</sup>* <sup>&</sup>lt; <sup>1</sup> 2 and the parameter *λ* is sufficiently small, then the generalized solution *u* belongs to the space *W*˚ <sup>2</sup> 2,*α*+*λ*−1(Ω) which is a subspace of *W*˚ <sup>2</sup> 2,*α*−1(Ω) (see [28]).

**Remark 2.** *Knowing that the solution belongs to the space W*˚ <sup>2</sup> 2,*α*+*λ*−1(Ω) *allows us to construct a finite element method for finding a generalized solution for the Dirichlet problem with the degeneration of the solution on the entire boundary of the domain with a convergence speed O*(*h*) *in the norm W*<sup>1</sup> 2,*α*(Ω)*.*
