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A boundary element Galerkin method for a hypersingular integral equation on open surfaces
Author(s) -
Ervin V. J.,
Stephan E. P.
Publication year - 1990
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670130402
Subject(s) - mathematics , galerkin method , piecewise , mathematical analysis , sobolev space , boundary (topology) , rate of convergence , bilinear form , integral equation , boundary element method , bilinear interpolation , gravitational singularity , finite element method , channel (broadcasting) , statistics , physics , electrical engineering , thermodynamics , engineering
A hypersingular boundary integral equation of the first kind on an open surface piece Γ is solved approximately using the Galerkin method. As boundary elements on rectangles we use continuous, piecewise bilinear functions which vanish on the boundary of Γ. We show how to compensate for the effect of the edge and corner singularities of the true solution of the integral equation by using an appropriately graded mesh and obtain the same convergence rate as for the case of a smooth solution. We also derive asymptotic error estimates in lower‐order Sobolev norms via the Aubin–Nitsche trick. Numerical experiments for the Galerkin method with piecewise linear functions on triangles demonstrate the effect of graded meshes and show experimental rates of convergence which underline the theoretical results.