MultiGridBarrier.jl

Quasi-optimal solvers for convex variational problems. MultiGridBarrier.jl solves nonlinear PDEs and boundary-value problems, including the nonsmooth ones that defeat most solvers: the p-Laplacian for every p ∈ [1, ∞], total variation, and obstacle problems. The multigrid barrier method couples an interior-point (barrier) method with a multigrid hierarchy to reach near-linear cost where that is provably achievable. Finite elements in 1D/2D/3D (simplicial P_k and tensor-product Q_k) and Chebyshev spectral discretizations, with optional CUDA GPU acceleration.

Install and quickstart

using Pkg; Pkg.add("MultiGridBarrier")

using MultiGridBarrier
geom = fem2d_P2()                                 # a 2D P2 triangular mesh
sol  = mgb_solve(assemble(amg(geom); p = 1.0))    # solve a (nonsmooth) p = 1 problem
plot(sol)

p = 1 is the hardest, nonsmooth case; larger p gives smoother problems. Swap fem2d_P2() for fem2d(), fem3d(), fem2d_P1(), spectral1d(), spectral2d(), or fem1d(; nodes).

Features

• Convex variational problems / nonlinear PDEs & BVPs: the p-Laplacian, total variation, obstacle-type constraints, and other convex functionals.
• Discretizations: finite elements in 1D/2D/3D (simplicial P1/P2 and tensor-product Q_k), plus Chebyshev spectral elements; all isoparametric.
• Solver: an algebraic-multigrid hierarchy (amg) driving a barrier (interior-point) method.
• Topological meshes: slit domains, branch cuts, and glued manifolds via explicit connectivity (the t= keyword and tensor_dofmap).
• GPU: optional CUDA acceleration.
• Time-dependent problems: parabolic_solve.

Documentation

Stable · Dev · Paper (PDF)

Bibliography

This package implements and builds on a growing line of work on barrier methods for convex problems in function spaces. If you use it in your research, please cite the paper(s) most relevant to your work:

• S. Loisel, The spectral barrier method to solve analytic convex optimization problems in function spaces, Numerische Mathematik 158(1):281–302, 2026. doi:10.1007/s00211-025-01508-0
• S. Loisel, Efficient algorithms for solving the p-Laplacian in polynomial time, Numerische Mathematik 146(2):369–400, 2020. doi:10.1007/s00211-020-01141-z

Author

Sébastien Loisel.