We present an a-posteriori error estimator for a finite element scheme applied to an elliptic boundary value problem with strong point singularities. The singularities caused by singular data are such that the problem is not well-posed in H1, but needs special weighted spaces for its variational formulation, see [1]. The corresponding finite element scheme uses standard piecewise polynomials but weighted with appropriate singular functions. The weight functions depend on the problem and the weighted energy space, see [2]. We define an a-posteriori error estimator for the special finite element scheme using a two-level hierarchical decomposition of the ansatz space. Our analysis follows Bank and Smith [3]. Main ingredient is a saturation assumption and a strengthened Cauchy-Schwarz inequality for the two-level decomposition in the weighted Sobolev spaces.

References

1. V.A. Rukavishnikov, On differentiability properties of an Rν-generalized solution of the Dirichlet problem. Dokl. Acad. Nauk SSSR, 309 (1989), no. 6, pp. 1318–1320; English transl. in Soviet Math. Dokl., 40 (1990), no. 3.

2. V.A. Rukavishnikov and H.I. Rukavishnikova, The finite element method for the third boundary value problem with strong singularity of solution. In: ENUMATH 97. Proceedings of the Second European Conference on Numerical Mathematics and Advanced Applications. World Scientific, Singapore, 1998, pp. 540– 548.

3. R.E. Bank and R.K. Smith, A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal., 30 (1993), no. 4, pp. 921- -935.