Nonuniform Error Estimates for the Finite Element Method

Error bounds that are not uniform in the solution u are obtained for finite element approximations to elliptic boundary value problems. The inequality \| u - uₕ \|₁ hs - 1 δ ₁ (h,u) is obtained, and where the error u - uₕ is measured in the norm of H¹ (Ω ), where for fixed u ∈ Hˢ (Ω ), δ ₁ (h,u) → 0 as h → 0. It is also shown that δ ₁ (h,u) cannot tend to 0 arbitrarily slowly.