A stable, implicit finite volume element (FVE) method for the steady state shallow ice approximation (SIA) free-boundary problem. Published as:
E. Bueler (2016). Stable finite volume element schemes for the shallow-ice approximation, J. Glaciol. 62 (232), 230-242, doi:10.1017/jog.2015.3.
We re-interpret the classical Mahaffy (1976) FD scheme as an FVE scheme. Then we construct an improved scheme. Both the classical and improved schemes use a 2D structured rectangular grid, piecewise-bilinear (i.e. Q1 finite element) trial function space, and a flux integral as the weak form. The improved scheme has better quadrature in the flux integral and a form of first-order upwinding which only acts on the bedrock-gradient part of the flux. We solve the steady-state problem by a continuation-modified and constrained Newton solver.
Directory paper/ contains the LaTeX sources and figures.
The C code in petsc/ implements both schemes using PETSc. Specifically, a 2d DMDA manages the structured grid SNESVI to solve the problem. The free boundary problem thus has the non-negative thickness constraint built-in. Examples include verification cases and two high-resolution Greenland set-ups.
This project lived inside my layer-conserve repo for much of its life.