109 problems

Uniform inf-sup conjecture. Based on empirical investigations of the inf-sup constant $\gamma^\delta$ for the first-order ultra-weak formulation, it is conjectured that

Integrability-relaxation conjecture. The integrability assumptions in the definition of a weak solution could potentially be relaxed. The paper reports convergence of the diffusion…

Let $\Omega$ be sufficiently smooth, let ${\bf p} \in [H^{k+1}(\Omega)]^2$, let $u \in H^{\max[3,k+1]}(\Omega)$, and let $\nabla \cdot {\bf p} \in H^{k+1}(\Omega)$. For the modifie…

Let $u$ be the exact solution and $u_h$ the numerical solution produced by the conservative flux optimization finite element scheme, with $\beta$, $\tau_D$, and $\boldsymbol{n}_e$…

Let $\Omega_c$ be the evolving domain in the clamped plate eigenvalue problem, and let $\lambda_{even}(c)$ and $\lambda_{odd}(c)$ denote the eigenvalues associated with even and od…

Let ${\mathcal X}$ and ${\mathcal Y}$ be subspaces of a Hilbert space ${\mathcal H}$ with ${{\rm dim}} \mathcal X={{\rm dim}} \mathcal Y$, and let $A$ be Hermitian. For a vector…

Let $\mathcal{T}$ range over triangulations of bounded polygonal domains in $\mathbb{R}^2$, let $\mathcal{T}$ have at most $N$ triangles, and let $P_{V(\mathcal{T})}$ denote the…

Let $u^2$ and $u^2_h$ denote the infinite-dimensional and discrete second iterates, respectively, and let $abla (u^2-u^2h)$ be measured in $L^2( Omega)$. Let $lambda^2h$ be the d…

Let $f$ be a differentiable reaction term whose derivative is discontinuous, and consider the finite element approximation experiment described in the source, in which the iterativ…

The extra-stabilised EVP is a low-order eigenvalue method for the Schrödinger eigenvalue problem (EVP), introduced in the paper and compared with other lower-bound methods in numer…

Consider the mixed finite element discretization of the vector Dirichlet Laplace boundary value problem in three dimensions on a convex domain. The current analysis gives an…

State-error convergence conjecture. The order of convergence in Lemma can be doubled, yielding an approximation of order

In Test-problem 3, let $\Omega=\Omega_{hol}$ be a hollow domain, with Neumann conditions prescribed on the outer boundary $\partial\Omega_1$ and Dirichlet conditions on the inner b…

Let the fine and coarse discretizations in the two-grid finite-element method for Helmholtz problems have finite-element spaces, with the coarse finite-element space not contained…

Estimator-behaviour conjecture. The estimator behaviour reported in Tables 7 and 8 of Gorynina, Lozinski, and Picasso is due to the different a posteriori error estimator used ther…

Consider the mixed-boundary-condition setting on a polyhedral subdomain $\omega_{\boldsymbol v}$, with the associated spaces…

Consider local Poincaré–Friedrichs inequalities on simplices, with boundary conditions imposed along some faces; in particular, compare the known improvements for triangles with th…

Let $T$ be a tetrahedron, and let $C_{{\rm PF},T,\Gamma,\ell,k,p}$ denote the Poincaré–Friedrichs constant for the relevant form degree, boundary face set, and Lebesgue exponent…

Stability conjecture. The stability property extends to three dimensions and to domains that are not topologically trivial.

Let $V$ be the Hilbert space, let $B\colon V\to L^2(\Omega_d;\mathbb{R}^m)$ be the observation operator, and let $u_h^k$ denote the proximal Galerkin iterates for the general probl…

Let $u_{0}$ be the solution of the homogenized equation, and let $u_{\mathcal{D}^{\text{p}}}^{\epsilon}$ be the solution of the partially dilation-dilated equation. Motivated by th…

Let $\rho(k)$ denote the stability constant for the Helmholtz problem at wavenumber $k$, and let $C$ and $k_0$ be positive constants. The best known general upper bound is…

Let $T^{\rm R}$ be the barycentric refinement of a $d$-dimensional simplex, let $\mathbb{S}$ denote the space of symmetric matrices, and let…

Directed-hyperarea vector identity. The directed-hyperarea vector satisfies

Optimization-challenge conjecture. When $beta$ is relatively small, so that the solution is smoother, a small $N$ can achieve good accuracy, whereas a larger $N$ may lead to greate…