HierarchicalProximalGalerkin

This repository implement the hierarchical proximal Galerkin solver (hpG) as described in

"Hierarchical proximal Galerkin: a fast $hp$-FEM solver for variational problems with pointwise inequality constraints", Ioannis. P. A. Papadopoulos (2024).

It combines the proximal Galerkin solver of Keith and Surowiec (paper) with the hierarchical p-FEM basis to achieve very-high-order discretizations of: (i) the obstacle problem, (ii) the generalized elastic-plastic torsion problem, and (iii) the thermoforming problem: an obstacle-type quasi-variational inequality.

Figure	File: examples/
1, 2, 3	spy-plots.jl
4	high-frequency/main_1d_high_freq.jl
5, 6	bessel_obstacle/main_oscillatory_obstacle.jl
7	gradient-bounds/main_gradient_bounds.jl
8	thermoforming.jl

Table	File: examples/
1	bessel_obstacle/main_oscillatory_obstacle.jl
2	gradient-bounds/main_gradient_bounds.jl
3	thermoforming.jl

Installation

The package is not registered. Please install via

pkg> add https://github.com/ioannisPApapadopoulos/HierarchicalProximalGalerkin.jl.git

Dependencies

This package relies on PiecewiseOrthogonalPolynomials.jl v0.5.1 for its implementation of the hierarchical p-FEM basis and FastTransforms.jl v0.16.5 that allow for fast quadrature.

It also utilizes SparseArrays.jl, MatrixFactorizations.jl, and IterativeSolvers.jl for fast linear algebra.

References

If you use this package please reference:

[1] Ioannis P. A. Papadopoulos. "Hierarchical proximal Galerkin: a fast $hp$-FEM solver for variational problems with pointwise inequality constraints." (2024).

[2] Brendan Keith and Thomas M. Surowiec. "Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints." Foundations of Computational Mathematics (2024): 1-97. DOI: 10.1007/s10208-024-09681-8

Contact

[email protected]