12 problems

Dispersion-error blending conjecture. The blendings

Spline approximation conjecture. The assumption is satisfied for all $C^{k_t}$ splines with polynomial degree $p_t\geq 2$ such that $p_t-k_t$ is odd. This would extend the stated v…

Quasi-optimal convergence conjecture. Based on numerical findings, for splines, quasi-optimal convergence occurs whenever the difference between degree and regularity is odd.

Quasi-optimal convergence conjecture. Based on numerical findings, for splines on uniform meshes, quasi-optimal convergence occurs whenever the difference between polynomial degree…

Let $\hat{K}_p$ and $\hat{M}_p$ be the stiffness and mass matrices for the $p$-th order softIGA method on the uniform mesh of $\Omega=[0,1]$ with $N$ elements, and let $T_p$ be the…

Let $\Omega_1$ be the dependent multi-patch domain, let $X^0_{h,0}\subset H^1_0(\Omega_1)$ be the discrete conforming scalar space, and let $K_h$ be the discrete kernel of the gaug…

Let the Laplacian have periodic boundary conditions, and let the Galerkin method use the periodic $n$-dimensional spline space of degree $p$ with maximal smoothness $C^{p-1}$ on a…

Let $\{\mathcal{M}^{\ell}\}_{\ell=0}^{\ell_{\max}}$ be the sequence of consecutively globally refined unstructured quad meshes described above, and let $\mathcal{B}^{\ell}$ be the…

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 $\overline{\Omega}\in\mathcal{A}$ be a randomly generated trilinear multi-patch volume with an inner edge of valency $\nu\in\{3,4,5\}$. Let $p$ be the spline degree, $r$ the re…

Let $\mathsf{A}$ be the matrix associated with the discretization under consideration, and let $\tilde{\mathsf{A}}$ be its surrogate matrix. Suppose that the entire kernel of…

Let ${\boldsymbol{l}}$ be a column of ${\boldsymbol{L}}$, and let $\bar{{\boldsymbol{H}}}$ be the sparse matrix constructed as its left null-space operator, with each column summin…