Weighted FEM operationalized for boundary value problems with singularity and inconsistent degeneracy of input data

Abstract

On the base of designed computational technologies, several comparative experiments and numerical analysis of the weighted finite element method based on the notion of R generalized solution and a classical finite element method are carried out. Boundary value problems for elliptic equations with singularity are divided into two classes: with consistent and inconsistent degeneracy of the input data. In boundary value problems with a consistent degeneration of input data, all components in differential equations have the same asymptotic behaviour in the neighbourhoods of singularity points. It means that the increasing order/degree of singularity arising in the equation terms when the derivative order grows is balanced out by the appropriate behaviour of coefficients. For computational solution of such problems, the concept of R -generalized solution is introduced and the finite element method (FEM) is developed. It allows the authors to find an approximate answer with a rate of O(h) with respect to the norm of a Sobolev weight space [1]. For boundary value problems with inconsistent degeneracy of the original data, all the coefficients of the equation have the same asymptotic order in the neighbourhoods of singularity points, and, it implies that all the terms of the equation have singularities of different order in these neighbourhoods. The simplest example of this problem class is the boundary value problems for differential equations and systems of equations in domains with a boundary containing reentrant angles. In [2], a special weighted set was allocated for such problems, in which it is possible to establish an existence and uniqueness of R -generalized solution. The weighted FEM designed by the authors [3] allowed them to define an approximate R -generalized solution without loss of accuracy and independent of the singularity size. The suggested computational technologies have been modified and developed for the issues of electromagnetism and hydrodynamics. For the system of Maxwell's equations, Stokes and Oseen's laws in domains with reentrant angles on the boundary, the weighted FEM exceeds in accuracy and utilization efficiency both the classical FEM and the FEM with mesh refinement to the singularity points [4-7]. In [8-11], this approach was developed for the problem of the theory of elasticity with singularity. Comparative analysis of many test problems found out that the values of absolute errors in the entire domain and in the neighbourhood of singularity points for the solution

Keywords

• 1 computational technologies
• boundary value problem with singularity
• weighted finite element method

1. Introduction

found by the weighted FEM are in two orders smaller than for the approximate solutions found by both classical FEM and FEM with mesh refinement.

This paper deals with several comparative experiments and numerical analysis of the accurate finding of approximate solution by the weighted FEM and classical FEM for test problems with different types of singularities. They are carried out for boundary value problems with inconsistent degeneracy of input data. Some conclusions are made about the efficient usage of the weighted FEM for finding solutions to boundary value problems with singularity.

2. Principal symbols. Problem statement

Assume that   R2 is a bounded domain with a piecewise-smooth boundary  and closure . Denote by  i , i =1, n , the cross points of the continuously differentiable boundary  and n  Oi = x : x − i   , and also Oi  Oj =  , i  j. Assume, ' = i=1i , where i =   Oi , i =1, n .

Let  (x) be a weight function defined as follows:  (x) =  ( x1 − x1(i) )2 + ( x2 − x2(i) )2 , x   ',

  , x   \  ', where ( x1(i) , x2(i) ) = i .

We introduce the weighted space W2k, () with squared norm u W2k, () =    2 Dmu 2 dx, 2

|m|k  where k is a nonnegative integer,  is a real nonnegative integer, Dm = |m| x1m1x2m2 , m = (m1, m2 ), | m |= m1 + m , mj , j = 1, 2 are nonnegative integers. If k = 0 , we will represent W20, () = L2, ().

2 By

W2k, +k−1(, ) (k = 1, 2,   0) denote a set of functions for which Dmu  c1 (  (x)) +|m| , x   ', where m = 0,1, 2, c1  0 is a constant independent on m , and u

L2, (\')

 c2  0, with a norm defined in ( 1 ).

Let W k2, (, ) be a subset of functions from the set W2k, (, ) which are going to zero almost everywhere on , and let Hk ,− (,c3 ) (k = 0,1) be a set of functions with a norm u Hk,− (,c3 ) = max ess sup  − +|m|Dmu  c3.

|m|k x If k = 0, then H0,− (,c3 ) = L,− (, c3 ).

The properties of the introduced weighted spaces and sets were studied in [9].

In the domain  , we consider the boundary value problem

2   u  −  akk (x)  + a(x)u = f (x), x , k=1 xk  xk 

u(x) = 0, x . ( 3 )

Definition 1. The boundary value problem ( 2 ), ( 3 ) will be called a Dirichlet problem with inconsistent degeneracy of input data, if the coefficients of the equation for some real number  satisfy the requirements akk (x)  H 1,− (, c4 ), a(x)  L,− (,c5 ) 2 2  akk (x)k2  c6  (x)k2 , k=1 k=1 ( 1 ) ( 2 ) ( 4 ) ( 5 ) f (x)  L2, (, ), where ci (i = 4,5, 6, 7) are positive constants that do not depend on x; 1,2  R,12 +22  0.

Introduce the bilinear and linear forms, respectively: a(u ,v) = 2  akk 2 u v  2 u v dx +  a 2 u v dx, k=1   xk xk + akk xk xk   l( ) =   2 f v dx.



Definition 2. A function u from the set W21, + /2 (, ) is called R -generalized solution of the problem ( 2 ), ( 3 ), if the identity u = 0 holds almost everywhere on  and for all v from a(x)  c7  (x) ( 6 ) almost everywhere on  , and the right hand side of the equation for some real nonnegative number  meets the condition ( 7 ) ( 8 ) W 12, + /2 (, ) the identity is valid for any fixed value  satisfying the inequality    +  / 2.

a(u , v) = l(v)

The membership of R -generalized solution to the weighted set W k2,+2+ /2+k+1(, ) was studied in [12, 14].

Remark 1. For boundary value problems with singularity caused by the degeneracy of input data (coefficients of a differential equation, right-hand sides of equation and boundary conditions), it is not always possible to determine a generalized solution. To suppress the singularity, a weight function  (x) is introduced into the bilinear and linear forms. The degree of it depends on the properties of the problem. The term of R -generalized solution is defined. This allows us to suppress singularity of the solution and provide convergence of integrals in the integral identity.

Remark 2. In [1] a Dirichlet problem with consistent degeneracy of input data is considered. For such a problem, all terms in bilinear form have the same order in neighbourhood of each singularity point. The difference of the problem inconsistent degeneracy of input data investigated in this paper is that the coefficients of equation ( 2 ) in the neighbourhood of singularity points  i , i = 1, n have the same asymptotic behaviour. That is why the additive components in a bilinear form have different order. Such feature of boundary value problems with inconsistent degeneracy of input data necessitates introducing of weighted set W21, + /2 (, ) , since there is a cluster of R -generalized solutions in a weighted space. The selection of a single R -generalized solution in the weighted set can be carried out by adjustment of parameters  and  ([14]).

3. Numerical experiments for a Dirichlet problem degeneracy of input data with inconsistent

In [3], a scheme of the weighted FEM is constructed, based on the definition of R -generalized solution of the problem ( 2 ), ( 3 ), the rate of convergence of the approximate solution of the suggested finite element method to the exact R -generalized solution in the weight set W21, + /2+1(, ) is investigated, and estimation of the finite-element approximation is assessed.

In this section, we provide numerical experiments and analysis of the obtained results for two test problems. A differential equation ( 2 ) is considered in test boundary value problems. The exact solution u(x) is chosen so that the boundary condition ( 3 ) are satisfied. Moreover, the coefficients akk (x), k = 1, 2, a(x) of the differential equation are chosen to satisfy conditions ( 4 ) to ( 7 ), and then the right hand side of the differential equation ( 2 ) is determined.

Numerical experiments implementing the weight FEM described above are carried out using computer program Proba-II and the GMRES method [15]. The optimal values of the parameters  and  are determined using the software package [16].

Calculations for each test problem are performed on grids with a different step h . The iteration process of solution of linear algebraic equations stops as soon as the norm of the difference between the approximate solutions on the last two iterations became smaller than 10−9 . For each test problem, both the approximate R -generalized solution and the approximate generalized solution uh ( = 0) were calculated. For the found approximate R -generalized solution, the error  was determined in the norm of the set W21, + /2+1(, )

 1/2  =     2( + /2+1) Dm (u − uh ) 2 dx  .

 |m|1  In each of the grid nodes Pi , i = 1, Nh , the absolute errors were determined for the approximate R generalized and generalized solutions

 (Pi ) = u(Pi ) − uh (Pi ) , (Pi ) = u(Pi ) − uh (Pi ) , i = 1, Nh , respectively, and then the values of the largest absolute errors were calculated  = im=1.a..Nxh  (Pi ),  = max (Pi ).

i=1...Nh We introduce the following notation: n1 – is the number of several sub-sections along the axes Ox1 and Ox2 ;  i , i = 1, 2 – is the specified limiting error; n2 – is the number of grid nodes where the absolute difference between the values of the exact and approximate generalized or R -generalized solutions exceeds the limiting error  1 ; n3 – is the number of grid nodes where the absolute difference between the values of the exact and approximate generalized or R -generalized solutions exceeds the limiting error  2 and is less than  1 ; n4 – is the number of grid nodes from the  -neighbourhood of the singularity point; Nit – is the number of iterations required to achieve requires accuracy; d – is the parameter used to calculate the radius of  -neighbourhood of the singularity point; hx1 – is the length of the partition segment along Ox1 axis;  = (1+ 0,01d)hx1 – is a radius of the neighbourhood of the singularity point;  – is the degree of the weight function in R -generalized solution.

Test problem 1. Suppose that

 = {x: (x1, x2 ), −1  x1  1, −1  x2  0}.

We choose the following function as an exact solution to test problem 1:

The equation coefficients are:

u(x) = ( x12 + x22 )2/3 sin cos (1 − x12 )(1 − x2 ).

a11(x) = a22 (x) = a(x) =

1 x12 + x22 .

Then the right-hand side of equation ( 2 ) takes the form:

1 f (x) = (−18x14 − 9x14 x2 + 9x14 x22 − 41x12 x2 + 56x12 x22 + 9x12 x24 +18x12 − 9

−9x12 x23 − 63x24 + 63x23 + 38x2 − 53x22 ) x1 / ( x12 + x22 )13/6 .

The exact solution u(x) of test problem 1 belongs to the sets W21,0 (, ) and W22,1/3 (, ) ; the coefficients akk (x), k = 1, 2 , and a(x) of the differential equation ( 2 ) belong to the sets H1,−1(,c8 ) а) 810−5  (Pi )  2 10−4 2 10−4  (Pi )  6 10−4 6 10−4  (Pi )  2 10−3 (Pi )  2 10−3 and L,−1(,c9 ) , respectively; the right-hand side f (x) of equation ( 2 ) belongs to L2,4/3(, ) . A weak singularity of the solution of this test problem is due to the degeneracy property of the coefficients akk (x), k = 1, 2 , and a(x) at the origin.

Numerical results for test problem 1 are presented in Tables 1-4 and on Figures 1-4.

Figure 1 shows distribution of the absolute errors of the approximate generalized and R generalized solutions, respectively. 8,86·10-3 following data: grid 128×64,  = 2, 2 ,  = 1 , 1 = 2 10−3 ,  2 = 6 10−4 . 2,58·10-2 1,73·10-2 8,86·10-3 2,53·10-3 σ 1,24·10-2 1,11·10-2 8,86·10-3 7,96·10-3

2 2,1 2,2 2,25 2,7 3 ν Figure 4: Influence of parameter  on accuracy of the approximate R -generalized solution, test problem 1.

Test problem 2. Suppose that For test problem 2,

 ={x: (x1,x2), −1 x1 1, −1 x2  0}. u(x) = ( x12 + x22 )−1/2 sin cos (1− x12)(1− x2), a11(x) = a22(x) =

1 1 , x12 + x22 , a(x) = x12 + x22 f (x) = −1(8x14 −4x12x22 x12 + x22 −53x12x22 −8x12 + 4x12x2 x12 + x22 +

4

The exact solution u(x) of the test problem 2 belongs to the sets W21,1/2 (, ) and W22,3/2 (, ) ; the coefficients akk (x), k = 1, 2 , and a(x) of the differential equation ( 2 ) belong to the sets H1,−1(, c10 ) and L,−2 (,c11) , respectively; the right-hand side f (x) of equation ( 2 ) belongs to L2,5/2 (, ) . The strong singularity of the solution of this test problem is due to degeneracy of coefficients akk (x), k = 1, 2 , and a(x) at the origin. A generalized solution to this problem does not exist, but R generalized solution can be defined.

Numerical results for test problem 2 are presented in Tables 5-7 and on Figures 5-8.

Figure 5 shows distribution of the absolute error of the approximate R -generalized solution in the domain. 0  (Pi ) 110−3 110−3  (Pi ) 110−2 110−2  (Pi )  310−2 310−2  (Pi )  9 10−2 (Pi )  9 10−2  = 2,1875 10−2 ) in domain  for n1 = 128 , test problem 2.

Table 5 and Figure 6 show dependence of accuracy of the approximate R -generalized solution on the grid size. Calculations were performed for test problem 2 with the following values:  = 6,5 ,  = 2 ,  = 2,1875 10−2 ( d = 10 ), 1 = 7 10−2 ,  2 = 310−2 . In test problem 2, an approximate generalized solution was not possible to find because of program failure.

Remark 3. Numerical results (Table 5) for test problem 2 show that the approximate R generalized solution can be found with high precision even when the generalized solution cannot be calculated. 80

128 1,38·10-5 7,62·10-6 2,26·10-6 8,02·10-2 8,5·10-2 9,87·10-2 1,01·10-4 1,38·10-5 2,26·10-6 performed for test problem 2 with the following data: grid 128×64,  = 6,5 ,  = 2 ,  1 = 7 10−2 ,  2 = 310−2 .

5,5 6 6,45 6,6 6,85 7 ν Figure 8: Influence of parameter  on accuracy of the approximate R -generalized solution, test problem 2.

Reviewing results of the numerical experiment, we can make the following conclusions about approximation properties of the weighted finite element method for boundary value problems with singularity of the solution and inconsistent degeneracy of input data:

1) The approximate R -generalized solution of the problem ( 2 ), ( 3 ) converges to the exact solution in the norm of the weighted set W21, + /2+1(, ) at a rate not less than O(h) (Table 1 and 5), which verifies the theoretical results obtained ([2]).

2) The introduction of the notion of R -generalized solution and application of the weighted finite element method allows us to deal with singularity caused by input data degeneracy (Table 5) and if there is no generalized solution. Although, an approximate R -generalized solution is highly precise even in the neighbourhood of the point of singularity.

3) For the best parameters  and  convergence rate of the approximate R -generalized solution to the exact one is the highest.

4. Acknowledgements

The reported study was supported by RSF according to the research project No. 21-11-00039.

5. References

problem with a singularity belongs to the space W2k,++2 /2+k+1(, ) , Differential Equations 45

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