An error estimate for the Laplace problem with edge singularities

다국어 초록 (Multilingual Abstract)

In a neighborhood of concave edge through the origin in a cylindrical region Q the solution of the Laplace equation with zero Dirichlet data can have a decomposition of the form u=ur+(c*E)Xr<SUP>π/ω</SUP> sin[(π/ω)(θ-ω₁)] with ur ∈ H²(Q) and c ∈ H<SUP>1-π/ω</SUP>(?) where ω = ω₂-ω₁ ＞ π is the opening angle, χ is a cutoff function and * is the convolution on the edge. We define the approximations ur,h and ch for the regular part ur and the edge flux intensity function c by applying the inverse Fourier transform to the finite element solution of the Laplace problem with parameter. We derive the error estimate ∥ur-ur,h∥I,Q+∥c-ch∥1-π/ω,R ≤ Ch<SUP>1-ε</SUP>∥f∥0,Q, where ε is an arbitrarily small positive number, which shows the rate of convergence faster than that expected by the regularity of the function c.

목차 (Table of Contents)

• ABSTRACT
• INTRODUCTION
• THE RESULT OF 2-DIMENSIONAL LAPLACE PROBLEM WITH PARAMETER
• MAIN RESULT
• REFERENCES