Compact Approximation of a Two-Dimensional Boundary Value Problem for Elliptic Equations of the Second Order with a Discontinuous Coefficient

Abstract

For an elliptic equation of the second order with variable discontinuous coefficients and the right side, a scheme of the fourth order of accuracy is constructed. On the jump line, the docking conditions (Kirchhoff) are assumed to be satisfied. The use of Richardson’s extrapolation, as the numerical experiments show, increases the order of accuracy to about the sixth order. It is shown that relaxation methods, including multigrid methods, are applicable to solve such systems of linear algebraic equations (SLAEs) corresponding to a compact finite-difference approximation of the problem. In comparison with the classical approximation, the accuracy increases by a factor of about 100 with the same complexity. Various variants of the equation and boundary conditions are considered, as well as the problem of determining the eigenvalues and functions for a piecewise constant coefficient of the equation.

APPENDIX

1.1 Bad Stencils for a Compact Approximation

In the main text of the article, “successful” stencils and coefficients of a compact approximation of the problem are presented. However, with other (sometimes more natural) stencils, implementation barriers can arise:

(i) When determining the coefficients of operators A and P, the corresponding SLAE of order N turns out to be degenerate.

(ii) The large matrix $\mathcal{A}$ turns out to be ill-conditioned. This usually manifests itself in violation of its property of negative definiteness.

We give examples of such pairs of stencils and indicate which of the two types of problem is observed.

(i) We choose the following stencils (see Figs. 2, 25): cross for the function u and square for the right side of f, i.e., ${{\nu }_{A}} = 5$, ${{\nu }_{P}} = 9$.

Fig. 25.

Cross-square stencil for a compact approximation on a square grid for the functions u and f, respectively.

Taking into account the symmetry of the problem, we obtain for an arbitrary internal node of a two-dimensional grid with numbers $i,j$ a linear algebraic equation with as yet undetermined coefficients $p,q,r$:

$$u(i,j - 1) + u(i - 1,j) + u(i + 1,j) + u(i,j + 1) - 4u(i,j)$$

$$ = r\left[ {f(i - 1,j - 1) + f(i - 1,j + 1) + f(i + 1,j - 1) + f(i + 1,j + 1)} \right]$$

(A.1)

$$ + \;q\left[ {f(i,j - 1) + f(i - 1,j) + f(i + 1,j) + f(i,j + 1)} \right] + pf(i,j).$$

To determine these coefficients, we require the exact fulfillment of Eq. (A.1) on the following pairs of test functions:

$$\begin{array}{*{20}{c}} {{\text{No}}{\text{.}}}&u&f&{{\text{Equation}}} \\ 1&1&0&{{\text{is fulfilled}}} \\ 2&{{{x}^{2}}}&2&{2{{h}^{2}} = 2[4r + 4q + p]} \\ 3&{{{x}^{4}}}&{12{{x}^{2}}}&{2{{h}^{4}} = 12{{h}^{2}}[4r + 2q]} \\ 4&{{{x}^{2}}{{y}^{2}}}&{2({{x}^{2}} + {{y}^{2}})}&{0 = 2[8r + 4q].} \end{array}$$

Equations (3) and (4) contradict each other. Therefore, in the specified stencil, a fourth-order scheme is impossible.

(ii) The case of degeneration or ill-conditioning of the global matrix $\mathcal{A}$. If we decide not to use the left and right limits of the right part of ${{f}_{ + }}$ and ${{f}_{ - }}$ on line Γ and assume that function f is not defined there (the stencil coefficients for the right side when approximating the equation there are 0), then it is not possible to choose a stencil for points on line Γ such that the fourth order of approximation would be provided and condition (9) would be satisfied. This circumstance often led to ill conditionality of the global matrix $\mathcal{A}$. An example of a “bad” stencil for points of type B is shown in Fig. 26. The test functions for this stencil are the same as in Eq. (4) and the normalization condition: a = 1. Solving the SLAE, we obtain the coefficients of the scheme, and $e1,e2,d1$, $d2 > 0$. Therefore, condition (9) is not satisfied. The numerical experiments indeed show the ill conditionality of the global matrix $\mathcal{A}$.

Fig. 26.

“Unsuccessful” stencils for type B points for solutions u and right side of f. Bold line, Γ.