Computational complexity of hierarchically adapted meshes

Abstract

We show that for meshes hierarchically adapted towards singularities there exists an order of variable elimination for direct solvers that will result in time complexity not worse than $$\mathcal {O}(\max (N, N^{3\frac{q-1}{q}}))$$, where N is the number of nodes and q is the dimensionality of the singularity. In particular, we show that this formula does not change depending on the spatial dimensionality of the mesh. We also show the relationship between the time complexity and the Kolmogorov dimension of the singularity.