Is there any Nonlinear case of hybridizable discontinuous Galerkin

Jim_Husman · May 18, 2025, 1:44am

Hi everyone ! Recently I am trying to solve my problem using hybridizable discontinuous Galerkin, However all the case I found are linear solver. Is there any Nonlinear case of hybridizable discontinuous Galerkin?

Thanks so much for any hint.

nate · May 19, 2025, 3:35pm

What i recommend below is a mess and likely will require a lot of effort to get it working again which I wouldn’t advise. But at least it could serve as a guide for your own superior implementation:

Here’s a nonlinear Poisson problem using HDG with legacy DOLFIN:

github.com/nate-sime/dolfin_dg

demo/dolfin/poisson/hdg_poisson_static_condensation.py

ca7ea9321

import numpy as np
from dolfin import (
    UnitSquareMesh, FunctionSpace, Function, TestFunction, dS, dx, errornorm,
    MPI, Expression, Constant, CellDiameter, FacetNormal, inner, grad, dot,
    div, DirichletBC, FiniteElement, MeshFunction, CompiledSubDomain,
    Measure)

import dolfin_dg.dolfin.hdg_newton
import dolfin_dg.hdg_form
import dolfin_dg.math

poly_o = 2
n_eles = [8, 16, 32, 64]
l2errors_u_l2 = np.zeros_like(n_eles, dtype=np.double)
l2errors_u_h1 = np.zeros_like(n_eles, dtype=np.double)
hs = np.zeros_like(n_eles, dtype=np.double)


for run_no, n_ele in enumerate(n_eles):

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Which uses this implementation of a Newton solver

github.com/nate-sime/dolfin_dg

dolfin_dg/dolfin/hdg_newton.py

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import leopart
import dolfin


def _formit(F):
    if isinstance(F, (list, tuple)):
        return list(map(_formit, F))
    return dolfin.Form(F)


class StaticCondensationNewtonSolver:

    def __init__(self, F, J, bcs,
                 rtol=1e-12, atol=1e-10, maximum_iterations=20,
                 krylov_solver=None):
        """
        Newton solver which implements static condensation via `leopart`. The
        residual and Jacobian are constructed as follows:

        F = [F_local, F_global]

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which in turn employs the LEoPart library (primary author @jmmaljaars) for static condensation of the cell-local problems in the HDG formulation

I would recommend the approach of dolfinx/python/demo/demo_static-condensation.py at main · FEniCS/dolfinx · GitHub for any future implementations of static condensation.

Both of these projects haven’t seen any attention since around 2021 as the primary contributors are now in industry.

BouteillerP · May 20, 2025, 7:10pm

You could also take a look at my repository where I’ve implemented a code to solve compressible Euler/Navier-Stokes equations using the HDG scheme:
https://github.com/PaulBouteiller/Hydra/tree/main
It’s still a work in progress, it’s not ready for use, and I plan to dedicate more time to it this summer. I hope you might find some useful insights there for your own problem. There is only one working demo on this repo namely the Sod shock test. The solver module relies on the blockNewton solver included in SCIFEM https://github.com/scientificcomputing/scifem and block manipulations from Nate’s github.

Jim_Husman · May 22, 2025, 6:27am

Thank you SO MUCH for you help !

Jim_Husman · May 22, 2025, 6:27am

Thank you SO MUCH for your help !

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Is there any Nonlinear case of hybridizable discontinuous Galerkin

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