A high-order accurate moving mesh finite element method for the radial Kohn--Sham equation

Abstract

In this paper, we introduce a highly accurate and efficient numerical solver for the radial Kohn--Sham equation. The equation is discretized using a high-order finite element method, with its performance further improved by incorporating a parameter-free moving mesh technique. This approach greatly reduces the number of elements required to achieve the desired precision. In practice, the mesh redistribution involves no more than three steps, ensuring the algorithm remains computationally efficient. Remarkably, with a maximum of $13$ elements, we successfully reproduce the NIST database results for elements with atomic numbers ranging from $1$ to $92$.

Figures (8)

• Figure 1: Iteration numbers for LOBPCG in solving the iron atom on a uniform mesh (top) and redistributed mesh (bottom). The left column displays the results without a preconditioner, while the right column shows the results with a preconditioner \ref{['eq:precon']}.
• Figure 2: Convergence results for iron atom on a uniform mesh with respect to different orders. Left is for the total energy and right is for the summation of the eigenvalues.
• Figure 3: Moving mesh method for the iron atom with order $p=4$.
• Figure 4: Meshes of iron atom for different number of elements.
• Figure 5: The number of elements of Fe atom in fixed and moving grids at different orders is compared.