A Note on the Convergence of Linear Finite Elements
Ricardo G. Durán · SIAM Journal on Numerical Analysis · 1988
Let u be the solution of the Dirichlet problem for Laplace’s equation in a smooth domain $\Omega $ and let $u_h $ be its piecewise linear finite element approximation. If the family of partitions is quasi-uniform it is known that \[ \left\| {u - u_h } \right\|_{L^p } \leqq Ch^2 \| u \|_{W^{2,p} } \quad {\text{for }}1 < p < \infty \] and \[\left\| {u - u_h } \right\|_{L^\infty } \leqq Ch^2 \log \frac{1} {h}\| u \|_{W^{2,\infty } } ,\] and moreover, the $\log ({1 / h})$ factor cannot be removed from the last estimate. The objective of this paper is to show that if $u \in W^{2,\infty } $, the error is of optimal order in the bounded mean oscillation (BMO) norm, that is, \[ \left\| {u - u_h } \right\|_{{\text{BMO}}} \leqq Ch^2 \| u \|_{W^{2,\infty } } .\] This norm is very close to the uniform norm, and moreover, the space BMO plays the role of $L^\infty $ in many applications in real analysis. The result is obtained by a duality argument as in the $L^p $ case, making use of an appropriate regularity result.