Boundary layer approximate approximations for the cubature of potentials.

*(English)*Zbl 0941.65023
Bonnet, M. (ed.) et al., Mathematical aspects of boundary element methods. Minisymposium during the IABEM 98 conference, dedicated to Vladimir Maz’ya on the occasion of his 60th birthday on 31st December 1997, Paris, France, 1998. Boca Raton, FL: Chapman & Hall/CRC. Chapman Hall/CRC Res. Notes Math. 414, 165-177 (2000).

From the introduction: We study a new method for the computation of multivariate integral operators with singular difference kernels over bounded domains
\[
{\mathcal K}u(x)= \int_\Omega g(x-y) u(y) dy.
\]
The accurate computation of such integrals and their derivatives is one of the main tasks, for example, in the solution of boundary value problems for partial differential equations with inhomogeneous right-hand side \(u\) by using boundary element methods.

Our approach is based on the concept of approximate approximations introduced by V. Maz’ya [A new approximation method and its applications to the calculation of volume potentials. Boundary point method. In: 3. DFG-Kolloqium des DFG-Forschungsschwerpunktes “Randelementmethoden” (1991)]. This concept offers new possibilities for high-order approximations of integral and pseudodifferential operators of mathematical physics. This is demonstrated by V. Maz’ya [MAFELAP ’93, 77-104 (1994; Zbl 0827.65083)] and by V. Maz’ya and G. Schmidt [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 6, No. 3, 161-184 (1995; Zbl 0845.68008)], where also the cubature error for integrals over the whole space is studied. The concept of approximate approximations was applied recently by the authors [Adv. Comput. Math. 10, No. 3-4, 311-342 (1999; Zbl 0935.65128)] to develop efficient cubature formulas for bounded Lipschitz domains. Here we consider a modification of this method to polyhedral domains leading to formulas of the same accuracy but of reduced complexity.

For the entire collection see [Zbl 0924.00038].

Our approach is based on the concept of approximate approximations introduced by V. Maz’ya [A new approximation method and its applications to the calculation of volume potentials. Boundary point method. In: 3. DFG-Kolloqium des DFG-Forschungsschwerpunktes “Randelementmethoden” (1991)]. This concept offers new possibilities for high-order approximations of integral and pseudodifferential operators of mathematical physics. This is demonstrated by V. Maz’ya [MAFELAP ’93, 77-104 (1994; Zbl 0827.65083)] and by V. Maz’ya and G. Schmidt [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 6, No. 3, 161-184 (1995; Zbl 0845.68008)], where also the cubature error for integrals over the whole space is studied. The concept of approximate approximations was applied recently by the authors [Adv. Comput. Math. 10, No. 3-4, 311-342 (1999; Zbl 0935.65128)] to develop efficient cubature formulas for bounded Lipschitz domains. Here we consider a modification of this method to polyhedral domains leading to formulas of the same accuracy but of reduced complexity.

For the entire collection see [Zbl 0924.00038].

##### MSC:

65D32 | Numerical quadrature and cubature formulas |

41A55 | Approximate quadratures |

41A63 | Multidimensional problems |

65N38 | Boundary element methods for boundary value problems involving PDEs |