Boundary element methods and their asymptotic convergence.

*(English)*Zbl 0618.65109
Theoretical acoustics and numerical techniques, CISM Courses Lect. 277, 135-216 (1983).

[For the entire collection see Zbl 0544.00039.]

Section 1 is devoted to strong ellipticity, Gårding’s inequality and the proof of Céa’s lemma along the lines of Michlin’s treatment of variational problems. As examples we present four different types of boundary integral equations. Three of them model the two and three- dimensional classical scattering problems in acoustics. These formulations are done in such a way that the boundary integral equations are perturbations of the case of wave number \(k=0\), i.e. the low frequency expansions are immediately available.

In section 2 we present the asymptotic error analysis for boundary element approximations in the framework of variational formulations in Sobolev spaces. In particular we prove the super approximation results for Galerkin’s method applied to strongly elliptic systems. Then we present recent results for ordinary collocation with odd degree splines for two-dimensional problems. We also review a comparison of the Galerkin method with collocation on curves and recent results on the collocation using even degree splines.

Section 3 deals with Fredholm integral equations of the second kind. First we apply our general approach obtaining superconvergence for smoothed approximations and for Galerkin’s solution at nodal points. Then we apply convergence results in Banach spaces to the special equations of classical scattering if the scatterer has corners and edges.

In section 4 we apply numerical integration to the foregoing procedures. For two-dimensional problems with the boundary equations on a curve we present the Galerkin collocation for equations with convolutional kernels in the principal part which combines the high accuracy of Galerkin’s method with the efficiency of ordinary collocation. The latter is also implemented in the same framework. For three-dimensional problems we present the isoparametric boundary element approximation by J. C. Nedelec [Comput. Meth. Appl. Mech. Eng. 8, 61-80 (1976; Zbl 0333.45015)]. The error estimates for the numerical quadratures are incorporated via the first Strang lemma leading to asymptotic errors for the fully discretized schemes.

Section 1 is devoted to strong ellipticity, Gårding’s inequality and the proof of Céa’s lemma along the lines of Michlin’s treatment of variational problems. As examples we present four different types of boundary integral equations. Three of them model the two and three- dimensional classical scattering problems in acoustics. These formulations are done in such a way that the boundary integral equations are perturbations of the case of wave number \(k=0\), i.e. the low frequency expansions are immediately available.

In section 2 we present the asymptotic error analysis for boundary element approximations in the framework of variational formulations in Sobolev spaces. In particular we prove the super approximation results for Galerkin’s method applied to strongly elliptic systems. Then we present recent results for ordinary collocation with odd degree splines for two-dimensional problems. We also review a comparison of the Galerkin method with collocation on curves and recent results on the collocation using even degree splines.

Section 3 deals with Fredholm integral equations of the second kind. First we apply our general approach obtaining superconvergence for smoothed approximations and for Galerkin’s solution at nodal points. Then we apply convergence results in Banach spaces to the special equations of classical scattering if the scatterer has corners and edges.

In section 4 we apply numerical integration to the foregoing procedures. For two-dimensional problems with the boundary equations on a curve we present the Galerkin collocation for equations with convolutional kernels in the principal part which combines the high accuracy of Galerkin’s method with the efficiency of ordinary collocation. The latter is also implemented in the same framework. For three-dimensional problems we present the isoparametric boundary element approximation by J. C. Nedelec [Comput. Meth. Appl. Mech. Eng. 8, 61-80 (1976; Zbl 0333.45015)]. The error estimates for the numerical quadratures are incorporated via the first Strang lemma leading to asymptotic errors for the fully discretized schemes.

##### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65R20 | Numerical methods for integral equations |

35J25 | Boundary value problems for second-order elliptic equations |

35C15 | Integral representations of solutions to PDEs |

45B05 | Fredholm integral equations |

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |