<i>A‐posteriori</i> error estimates for the finite element method

TLDR

The approach resembles the residual method but uses norms of negative Sobolev spaces tied to the bilinear energy form. The study derives computable a‑posteriori error estimates for finite element solutions in an asymptotic form as the mesh size h tends to zero. These estimates are obtained for one‑dimensional model problems—including source, eigenvalue, and parabolic equations with linear self‑adjoint second‑order operators—and can be extended to higher dimensions with additional considerations. The resulting estimates enable practical a‑posteriori accuracy assessment and form the basis for designing adaptive finite element solvers.

Abstract

Abstract Computable a‐posteriori error estimates for finite element solutions are derived in an asymptotic form for h → 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one‐dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self‐adjoint operator of the second order. Generalizations to more general one‐dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a‐posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.