Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes

In [10, 11] D. Yu proved that on a uniform square grid the error of the finite element method is controlled by either the element residuals or the edge residuals, i.e. jumps of the normal derivative across interelement boundaries, depending on whether the degree of the finite element functions is even or odd. The proof strongly relies on the uniformity of the grid, the tensor-product structure of the finite element space, and symmetry properties of the Legendre polynomials. A similar result for finite element methods on triangular grids is not known to us. Although it is often claimed that the edge residuals dominate the error of linear finite elements, a rigorous proof of this is also not known to us. In what follows we will prove that, up to higher order perturbation terms, the edge residuals yield global upper and local lower bounds on the error of linear finite element methods both in Hand L-norms. The proof of the lower bound follows from the results in [6]. It only needs shape regularity of the triangulation. The proof of the upper bound is based on the stability with respect to a suitable mesh-dependent norm of the L-projection onto linear