MiS Preprint Repository

In this article, we develop and analyze error estimators for a general class of second-order linear elliptic boundary value problems in bounded three-dimensional domains. We first describe the target class of problems, and assemble some basic mathematical facts and tools.

We then briefly examine discretizations based on tetrahedral partitions and conforming finite element subspaces, introduce notation, and subsequently define an error estimator based on the use of piecewise cubic face-bump functions that satisfy a residual equation.

We show that this type of indicator automatically satisfies a global lower bound inequality thereby giving efficiency, without regularity assumptions beyond those giving well-posedness of the continuous and discrete problems.

The main focus of the paper is then to establish the reverse inequality: a global upper bound on the error in terms of the error estimate (plus an oscillation term), again without regularity assumptions, thereby giving also reliability. To prove this result, we first derive some basic geometrical identities for conforming discretizations based on tetrahedral partitions, and then develop some interpolation results together with a collection of scale-invariant inequalities for the residual that are critical for establishing the global upper bound.

After establishing the main result, we give an analysis of the computational costs of actually computing the error indicator. Through a sequence of spectral equivalence inequalities, we show that the cost to evaluate the indicator (involving the solution of a linear system) is linear in the number of degrees of freedom.

We finish the article with a sequence of numerical experiments to illustrate the behavior predicted by the theoretical results, including: a Poisson problem on a 3D L-shaped domain, a jump coefficient problem in a cube, a convection-diffusion problem, and a strongly anisotropic diffusion problem.