Extend solenoidal projection stability beyond Duran’s setting

Establish the uniform $L^p$ stability estimate for the solenoidal projection of the Raviart–Thomas mixed method beyond Duran’s setting, including polygonal domains, general adaptive meshes, and discontinuous anisotropic coefficients.

Background

The LpL^p flux-error analysis depends on Hypothesis (SP), which requires an hh-independent bound for the weighted projection onto the discrete divergence-free Raviart–Thomas subspace. The paper explains that the available result of Duran applies only under restrictive assumptions: a smooth simply connected domain, quasi-uniform triangulations, and a Lipschitz scalar coefficient bounded away from zero.

The configurations of primary interest in the paper involve polygonal domains, fitted or adaptive meshes, and discontinuous anisotropic coefficients. The authors therefore retain the projection estimate as a hypothesis in these settings; proving it would extend the quasi-best approximation and subsequent flux-error theory to the broader class of problems studied numerically.

References

We are not aware of any result giving eq:Duran-stability on a polygon, on general adaptive meshes, or for the discontinuous and anisotropic coefficients considered here. Outside Duran's setting we keep it as a hypothesis.

eq:Duran-stability:

∥Rh∥Lp()≤CSP∥∥Lp()∀ ∈Lp()2.\|R_h\|_{L^p()}\le C_{\mathrm{SP}}\|\|_{L^p()} \qquad\forall\,\in L^p()^2 .

— Mixed Finite Element Methods for a Dirac Source: Divergence-Form Splitting and L^p Error Analysis  (2608.17575 - Wu et al., 18 Aug 2026) in Remark [Scope of (SP) and (SP′)], Section 4.2