8 problems

Let $\Omega\subset\mathbb{R}^3$ be a domain equipped with a uniform three-dimensional Freudenthal tetrahedral mesh $\mathcal{T}_h$. For every integer velocity degree $k\ge 4$, let…

For every nondegenerate tetrahedron $T_0\subset\mathbb{R}^3$ and every resolution of ties between equally long edges, repeatedly bisecting a longest edge produces a family…

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…

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

Let $T$ be a triangle with maximal interior angle $\omega$, and let $C_{\rm st,2}$ denote the stability constant associated with the hybrid high-order method. For any…

Let $\varphi_h\in\widetilde{\mathcal{V}}^1_{h,0}$ solve the discrete problem on the two-patch domain, and let $p_1^{(S)}\geq 2$, $p_2\geq 3$, and $r_1,r_2,\widetilde{r}\geq 1$. Def…

Let $\Omega$ be the spatial domain, $h$ the mesh size, and $k$ the polynomial order in the CGDG discretization. Let $e_w$, $\bm{e}_\theta$, $\bm{E}_\kappa$, and $\bm{e}_\gamma$ den…

Let $H=L^{1}([x_{\min},x_{\max}],\mathbb{R}_{+})$, let $H^{N}$ be the piecewise-constant approximation space generated by the basis functions $\beta_i^N$, and let $f^{N}(t,x)$ solve…