Background I am studying a numerical method for solving 2D heat transfer problem (a splitting scheme in time combined with certain space approximation using FEM with rectangular meshes). On the uniform mesh (as a certain simplified case) using the derived eigenvectors and eigenvalues of the arising discrete operators, the following condition was obtained as necessary for convergence:
Statement Consider a function $T^0(x,y)$ (the initial temperature) which can be decomposed after projecting to the uniform mesh with nodes $(x_i, y_j)$ as: $$ T^0(x_i,y_j) = \sum_{i=1}^{N} \sum_{j=1}^{N} T_{ij} \, u_i(k) \, u_j(l) $$ where $u_i, u_j$ are ``discrete harmonics'' (eigenvectors) $$ u_i(k) = sin \frac{\pi i(2k-1)}{2N}, i, k = 1, ... N. $$ and $T_{ij}$ - decomposition coefficients.
Then for convergence of the method we need the following sum to be bounded uniformly w.r.t mesh step $h$: $$ \sum_{i=1}^{N} \sum_{j=1}^{N} T^2_{ij} \lambda_j^2 (\lambda_i + \lambda_j)^3 \leq C < \infty $$ with $C$ independent from $h$ and $N$.
Here $h = 1/N \rightarrow 0$, $\lambda_k \sim \frac{\gamma_k}{h^2}$, $\gamma_k = 2 sin \frac{\pi k h}{2}$, $k = 1, ..., N$.
My question is whether the condition above (or any similar) is known in the literature and/or there are any results related to the class of functions which satisfy this condition?
The simplified version of the condition (taking $\gamma^k \sim \pi k h$, $h \rightarrow 0$) reads as follows: $$ \sum_{i=1}^{N} \sum_{j=1}^{N} T^2_{ij} j^4 (i^2 + j^2)^3 \leq C < \infty $$ with $C$ independent from $h$ and $N$.
Any remarks and comments are welcome.
