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On applications and the convergence of boundary integral methods. (English) Zbl 0561.65085
Treatment of integral equations by numerical methods, Proc. Symp., Durham 1982, 463-476 (1982).
[For the entire collection see Zbl 0499.00015.]
We consider the class of strongly elliptic boundary integral equations which provide coerciveness in form of Gårding inequalities. In special cases coercivity can also be obtained with potential theoretic methods but we prefer the more general approach via pseudodifferential operators. For the numerical treatment of these equations we formulate asymptotic error estimates in terms of the diminishing mesh width h of families of boundary elements. These asymptotic error estimates are in rather good agreement with numerical experiments and can also be used for making decisions concerning the choice of numerical quadrature formulas or the choice of orders of finite element spaces approximating the boundary and the desired charges. We give a brief survey of the asymptotic error results. For strongly elliptic equations we have quasi- optimality for Galerkin’s method in the energy space and superapproximation due to Aubin-Nitsche duality. In two dimensions, i.e. equations on curves we present corresponding results for collocation methods with odd degree splines and even degree splines. In the last section we present four different types of strongly elliptic boundary integral equations belonging to engineering applications.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations