The \(hp\)-version of the boundary element method on polygons. (English) Zbl 0888.65121
The authors give a new proof for the exponential convergence of the \(hp\)-version of the boundary element method for the Symm’s integral equation with logarithmic kernel, and for the hypersingular integral equation resulting from taking the normal derivative of the double layer potential. The approach is based on the asymptotic expansions of the solutions in singularity functions near the vertices of the polygonal boundary. This allows to show that the solutions belong to countably normed spaces; hence the solutions can be approximated exponentially fast in the energy norm by using the \(hp\) Galerkin solutions. The results are applied to the transmission problem for acoustic scattering of time-harmonic waves, and to the two-dimensional crack problems in linear elasticity.
Reviewer: O.Titow (Berlin)
MSC:
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 65R20 | Numerical methods for integral equations |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 76Q05 | Hydro- and aero-acoustics |
| 76M15 | Boundary element methods applied to problems in fluid mechanics |
| 74S15 | Boundary element methods applied to problems in solid mechanics |
| 74R99 | Fracture and damage |
Keywords:
exponential convergence; boundary element method; Symm’s integral equation; logarithmic kernel; hypersingular integral equation; asymptotic expansions; singularity functions; countably normed spaces; energy norm; Galerkin solutions; acoustic scattering; two-dimensional crack; linear elasticityReferences:
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