14 problems

Stationary-state conjecture. The discrete stationary states are governed by $p=0$ and some non-standard discretization of the velocity divergence.

Let $p$ denote the polynomial degree of a discontinuous Galerkin (DG) spatial discretization, and consider its dispersion-dissipation relations for a linear hyperbolic problem. Hu–…

Let $D$ be an operator, let $V$ be the ambient space, and let $a(\cdot,\cdot)$ be a bilinear operator. Define … where $\epsilon>0$ is small. Suppose that $a(u,v)$ is coercive on…

Roos and Zarin's extension conjecture. The analysis of uniform supercloseness for the nonsymmetric discontinuous Galerkin method with interior penalties can be extended from proble…

Robustness conjecture. The documented differences in robustness are due to the presence of both small-scale under-resolved features and significant variations in the density.

Let $\Omega$ be the domain, let $\mathcal{T}_h$ be a proper mesh in dimension $d$, and let $u$ be the exact solution of the model problem with $u\in H^{k+1}(\Omega)$, while $u_h$ i…

Graded-mesh reconstruction-error conjecture. In general, provided $r\ge2$,

Superconvergence conjecture. For $r\ge2$, there is a constant $C$ such that

Dissipation-balancing conjecture. Choosing a seemingly less dissipative convection term is simply counterbalanced by other numerical dissipation mechanisms.

The algorithm uses $N_{Pic}$ Picard iterations, typically with $N_{Pic}\in\{p_{\gamma}+1,p_{\gamma}+2\}$, where $p_{\gamma}$ is the polynomial degree in time; for ordinary differen…

Let a nodal basis be a basis of polynomial functions associated with prescribed interpolation points on an element, and call it orthogonal when its basis functions are pairwise ort…

Smoothness conjecture. The smoothness of the filtered discontinuous Galerkin computation is the same as the smoothness in the domain interior where the symmetric filter is applied.

Boundary Korn inequality. There exists a positive constant $\tilde{K}$ such that, for every $\mathbf{u}\in\mathbf{H}^2(\Omega)$,

Mesh-grading conjecture. The condition