Quadrature Formula for the Direct Value of the Normal Derivative of the Single Layer Potential
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RICAL METHODS
Quadrature Formula for the Direct Value of the Normal Derivative of the Single Layer Potential P. A. Krutitskii1∗ , I. O. Reznichenko2∗∗ , and V. V. Kolybasova2∗∗∗ 1
Keldysh Institute of Applied Mathematics, Moscow, 125047 Russia 2 Lomonosov Moscow State University, Moscow, 119991 Russia ∗ e-mail: [email protected], ∗∗ [email protected], ∗∗∗ [email protected] Received February 26, 2020; revised February 26, 2020; accepted May 14, 2020
Abstract—A quadrature formula for the direct value of the normal derivative of the single layer potential with smooth density defined on a closed or nonclosed surface is obtained. Single layer potentials for the Laplace and Helmholtz equations are considered. Numerical tests confirm that our quadrature formula gives considerably higher accuracy than the standard quadrature formula. It can be used when numerically solving boundary value problems for the Laplace and Helmholtz equations by the potential method and the boundary integral equation method. DOI: 10.1134/S001226612009013X
INTRODUCTION The standard quadrature formulas used in engineering calculations of the single layer potential for the Laplace and Helmholtz equations do not provide a uniform approximation to the potential near the surface Γ on which the density of the potential is defined and even tend to infinity as the point at which the quadrature formula is calculated approaches certain points of Γ [1, Ch. 2], even though the potential itself is continuous in the entire space, including all points of Γ. Consequently, the standard quadrature formulas do not preserve the potential’s very important property of being bounded and continuous on Γ. An improved quadrature formula preserving this property of the single layer potential was proposed in [2]. The present paper uses the approach in [2] to obtain an improved quadrature formula for the direct value of the normal derivative of the single layer potential on Γ. In particular, such a formula can be used for the numerical solution of the integral equations that arise when solving boundary value problems for the Laplace and Helmholtz equations by the potential method. 1. STATEMENT OF THE PROBLEM Consider the Cartesian coordinate system x = (x1 , x2 , x3 ) ∈ R3 in the Euclidean space R3 . Let Γ be a simple smooth closed or bounded nonclosed surface of the class C 2 containing the limit points of itself [3, Ch. 14, Sec. 1]. If the surface Γ is closed, then it must bound a volume–simply-connected interior domain [4, p. 201]. Assume that Γ is parametrized in such a way that the following rectangle is mapped onto it: y = (y1 , y2 , y3 ) ∈ Γ, y1 = y1 (u, v), y2 = y2 (u, v), y3 = y3 (u, v); u ∈ [0, A], v ∈ [0, B]; yj (u, v) ∈ C 2 ([0, A] × [0, B]), j = 1, 2, 3.
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The sphere, the surface of an ellipsoid, the smooth surfaces of solids of revolution, and many other, more sophisticated surfaces can be parametrized in such a way. In the rectangle [0, A] × [0, B], we partition the side [0, A] into N equal segments and the side [0, B] into M equal segments
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