Precise finite element error analysis by Yamamoto's explicit inversion formula for tridiagonal matrices -- an extension of Babuška-Osborn's theorems
The paper is concerned with error estimates for linear spline approximations \(u_h\) to boundary value problems of the form \(-(pu')'= f\) on \((a, b)\), \(u(a)= u(b)= 0\), with \(p\in L^\infty(a, b)\) bounded away from zero. The error analysis is based on an explicit formula for the inverse of the (tridiagonal) discretized matrix, and a splitting of the difference between this inverse and the Green's function into an approximation and an interpolation error term, resulting in a corresponding splitting of the error \(u -u_h\). Particular emphasis is put on the question if, for a sufficiently ``rough'' coefficient function \(p\), the convergence \(u- u_h\to 0\) can be arbitrarily slow, which has been answered affirmatively by \textit{I. Babuska} and \textit{J. Osborn} [Math. Comput. 69, No. 230, 443--462 (2000; Zbl 0940.65086)] for a more specific boundary value problem. Using their construction of \(p\), the author proves that the same holds true for the more general boundary value problem under consideration here, by detailed estimates of approximation and interpolation error.
scientific article; zbMATH DE number 1967711
scientific article