Anisotropic mesh adaptation for the Poisson equation

The goal of the code is to solve the Poisson equation

$$ \begin{cases} -\Delta u &= f, \text{ in } \Omega,\\ u &= g, \text{ on } \partial \Omega. \end{cases} $$

using the finite element method, with anisotropic mesh adaptation, in a (yet) two dimensional domain $\Omega\subset\mathbb R^2$. The methods used in this code are described in Picasso, 2003¹ and Dubuis, 2019². The main novelty of this code is the use of the open source softwares Fenicsx³ and mmg⁴. The code also allows to use $\mathbb P^2$ finite elements, using Naga-Zhang post-processing⁵. We provide a list of the file in this repository and what they contain.

• adaptive_algorithm.py : contains the class Params used to as an input of the adaptive solver solver_adapt.
• compute_error.py : gives the compute_error function, which computes the error between a finite element solution and a given known solution $u$.
• estimators.py : Contains the implementation of the anisotropic residual based a posteriori estimators that are used to drive the mesh adaptation. Contains also many functions used for the mesh adaptation, as well as the call to mmg.
• mesh_functions.py : Auxiliary functions used for the post-processing in ZienkiewiczZhu.py, NagaZhang_P1.py and NagaZhang_P2.py.
• NagaZhang_P1.py, NagaZhang_P2.py : Naga-Zhang post-processing⁵ of the gradient, for $\mathbb P^1$ or $\mathbb P^2$ finite elements.
• read_functions.py : A file that contains functions to read the data saved from the folder.
• run_solver.py : An example run of the adaptive algorithm, on the function (citer George).
• solver.py : A simple solver of the Poisson equation using Fenicsx³.
• ZienkiewiczZhu.py : Python implementation of the Zienkiewicz-Zhu post-processing method⁶ for $\mathbb P^1$ finite elements.

Footnotes

1. An Anisotropic Error Indicator Based on Zienkiewicz--Zhu Error Estimator: Application to Elliptic and Parabolic Problems ↩

2. Adaptive algorithms for two fluids flows with anisotropic finite elements and order two time discretizations ↩

3. dolfinx ↩ ↩²

4. mmg ↩

5. A New Finite Element Gradient Recovery Method: Superconvergence Property, A Posteriori Error Estimates Based on the Polynomial Preserving Recovery, The polynomial-preserving recovery for higher order finite element methods in 2D and 3D ↩ ↩²

6. The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity ↩