57 problems

Equivalence conjecture. The quantity $R_T$ defined by equation (def-RT) and the projected circumradius of $T$ are equivalent.

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain, let a positive piecewise constant function be given on a decomposition of $\Omega$ into subdomains, and let $d$ denot…

Let method denote the global gradient-taming method considered in the paper, and let the spatial discretization be defined on a domain. Global gradient-taming suitability conjectur…

Let the FEM and FD schemes be the finite-element and finite-difference numerical approaches used for the salt-crystallization problem described in the paper. FEM convergence-rate c…

Polar target-space conjecture. Since neither improved convergence rates for the enriched formulation $\mathcal{GE}^{2+}(V)$ nor oscillations for the $\mathcal{GE}^{2}(V)$ formulati…

Baumann–Sauter conjecture. The Galerkin method applied to $A_k$ and $A'_k$ does not suffer from the pollution effect.

Enhanced finite element conjecture. Using such constructions should have the ability to remove the Lavrentiev gap, thereby enabling the use of standard finite elements for these ty…

A degree-$p$ TPSS-SBP operator is exact for polynomials up to degree $p$, and is therefore expected to yield a solution convergence rate of $p+1$ when used to discretize PDEs with…

Let the bedrock elevation and surface mass balance be fixed over a land region, and consider the continuous-space, implicit time-step free-boundary problem for the glacier surface…

Let $c_{\rm eff}$ be the local efficiency constant in the estimate … where $\eta_\nu$ is the vertex error estimator and $\mathcal E(\omega^\nu)$ is the corresponding local error me…

p-independence conjecture. For general meshes, the dependence of $C_{{\rm stab},\nu}$ on $p$ is artificial and can be removed, so that the stability estimates hold with a constant…

Arbitrary-dimensional finite element construction conjecture. The $C^r$ finite elements can be constructed with this local shape function space and these degrees of freedom.

Let $\W$ be the domain, let $n$ denote its dimension, and let $q$ be the integrability exponent. An approximation is bilateral when it preserves both the lower and upper bounds, an…

Let $g$ be a smooth metric, let $\theta$ be a tensor field with the regularity assumed in Theorem, and let $B$ belong to the test-tensor space $\mathring{\mathcal{B}}$. The quantit…

Boundary-discretization conjecture. For $n=4$, the error in the $H^1$ semi-inner product is significantly worse than in the other two examples because this level of boundary discre…

Let $u$ be the stray-field potential on ${\mathbb R}^3$, and let $\boldsymbol M$ be the magnetization in a domain $\Omega$. Assume that $u\in W^2_2({\mathbb R}^3)$; equivalently, a…

Let $V_h$ be the continuous piecewise-polynomial velocity space of degree $4$ on the structured Freudenthal tetrahedral meshes, and let $\Sigma_h$ denote the associated potential s…

Let $V_h\subset[H^1(\Omega)]^3$ be constructed with continuous Lagrange elements of degree $k$. Let $\Sigma_h\subset H^1(\operatorname{curl},\Omega)$ be the space preceding $V_h$ i…

Asymptotic behavior conjecture. As $h\searrow 0$, $\|\nabla z_h\|_0^2$ will asymptotically behave as

Let $u$ be the exact solution of the two-dimensional vector Laplacian problem and $u_h$ its approximation by the order-$k$ nonconforming primal hybrid finite element method. Let…

Let the two-dimensional vector Laplacian be discretized by the quadratic nonconforming finite element developed by Brenner and Süli for the problem described above. Brunet–Süli con…

Let a family of three-dimensional Worsey–Farin splits be obtained from a family of shape-regular simplicial meshes. Shape regularity conjecture. The resulting family of Worsey–Fari…

Finite-element locality conjecture. Minimizing $W_{\mathrm{magic}}^+$ in a finite element space fails to find microstructure.

Orthogonality conjecture. The observed behavior is conjectured to occur because the condition $\nabla\cdot u=0$ is also required to be orthogonal to the pressure space.

Crouzeix–Falk's conjecture. The pair