Nonconforming methods and Strang lemmas

Let the exact solution lie in a continuous space V and the discrete solution in V_h, which need not be a subspace of V. Assume a genuine norm is defined on their sum, or on a suitable regularity-enriched sum containing the exact solution. The discrete bilinear form must extend to the pairs used in the analysis and be bounded there. Without that extension, writing a_h(u,w_h) can be meaningless even if the discrete matrix itself is perfectly well defined.

A broken gradient seminorm alone is not a norm on arbitrary discontinuous cellwise functions: every piecewise constant has zero gradient on each cell. Nonconforming boundary constraints, face-average matching, or jump penalties must remove or control the relevant kernel. Choose a norm and prove uniform discrete coercivity in it. These details distinguish a valid nonconforming approximation from an unsupported attempt to copy the conforming proof using elementwise derivatives and no control of jumps.

Suppose a_h is coercive with constant alpha_h on V_h and bounded with constant M_h when its first argument is an error from the common sum space and its second is discrete. Compare u_h with an arbitrary v_h. Coercivity bounds their difference by the approximation mismatch u−v_h and the consistency residual, divided by alpha_h. The triangle inequality adds the original approximation term. Taking the best candidate yields the displayed sufficient version of the second Strang estimate.

For a conforming method with perturbed quadrature, one can instead organize the residual into bilinear-form and load perturbations; the first Strang lemma tracks those variational crimes explicitly. Either organization needs uniform stability, meaningful continuity bounds, approximation tending to zero, and consistency defects tending to zero in the relevant norms. A small algebraic residual for the perturbed system does not ensure that its solution approaches the unperturbed weak solution.

For Crouzeix–Raviart affine elements, neighboring traces need agree only in their face averages, with corresponding homogeneous boundary averages. Elementwise integration by parts expresses the consistency defect in terms of exact normal fluxes paired with jumps. The zero-average jump cancels constant flux components, and a regularity estimate controls the remainder. This explains how a nonconforming space can converge, but it is not permission to ignore the face terms or to assume a rate without regularity and mesh hypotheses.

The same framework reveals why a fixed multiplicative stiffness error need not disappear on a finer mesh. If the assembled energy remains ten percent too large while the load stays exact, increasingly accurate spaces approximate a different variational problem. Solver tolerances and quadrature accuracy therefore need to be compatible with the desired discretization accuracy. When domains or boundary data are approximated, their contribution belongs in the same explicit consistency and data-error accounting rather than being hidden inside the mesh label.