Residual-based a posteriori error estimate for hypersingular equation on surfaces

Summary.The hypersingular integral equation of the first kind equivalently describes screen and Neumann problems on an open surface piece. The paper establishes a computable upper error bound for its Galerkin approximation and so motivates adaptive mesh refining algorithms. Numerical experiments for triangular elements on a screen provide empirical evidence of the superiority of adapted over uniform mesh-refining. The numerical realisation requires the evaluation of the hypersingular integral operator at a source point; this and other details on the algorithm are included.

[1]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[2]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[3]  E. P. Stephan,et al.  The $h-p$ version of the boundary element method on polygonal domains with quasiuniform meshes , 1991 .

[4]  Carsten Carstensen,et al.  Numerische Mathematik A posteriori error estimate and h-adaptive algorithm on surfaces for Symm ’ s integral equation , 2001 .

[5]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM , 2002, Math. Comput..

[6]  Ernst P. Stephan,et al.  An adaptive two-level method for hypersingular integral equations in \(R^3\) , 2000 .

[7]  Carsten Carstensen,et al.  Efficiency of a posteriori BEM-error estimates for first-kind integral equations on quasi-uniform meshes , 1996, Math. Comput..

[8]  Ernst P. Stephan,et al.  Regularity of mixed boundary value problems in ℝ3 and boundary element methods on graded meshes , 1990 .

[9]  Christian Lage,et al.  Transformation of hypersingular integrals and black-box cubature , 2001, Math. Comput..

[10]  Ricardo H. Nochetto,et al.  Small data oscillation implies the saturation assumption , 2002, Numerische Mathematik.

[11]  Carsten Carstensen,et al.  An a posteriori error estimate for a first-kind integral equation , 1997, Math. Comput..

[12]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM , 2002, Math. Comput..

[13]  J. Nédélec,et al.  Integral equations with non integrable kernels , 1982 .

[14]  Carsten Carstensen,et al.  A posteriori error estimates for boundary element methods , 1995 .