Explicit a priori error estimation for FEM solutions

Xuefeng LIU (Last updated: 2020/05/13)

Problem description

We consider the homogeneous boundary value problem of the Poisson equation over a bounded 2D domain.

$$ -\Delta u = f \quad \mbox{ in } \Omega; \quad u=0 \mbox{ on } \partial\Omega. $$

Let $u_h$ be the linear conforming FEM solution to the above problem. We provide an a priori error estimation to $u_h$ with an explicit value of $C_h$ such that $$ |\nabla(u-u_h)| \le C_h |f| \quad \forall f \in L^2(\Omega)\:. $$

The hypercircle method is adopted here. See detail in https://doi.org/10.1137/120878446 .

Codes

• create_mat.ipynb: The python code to create the matrices.
• a_priori_error_estimation.ipynb: Calculate the quantity $\kappa_h$ and the explicit value (upper bound) of $C_h$. Language: Octave.

Environment.

• Python3 + FEniCS for create_matrix.ipynb .
• Matlab or Octave for a_priori_error_estimation.ipynb .

Setting of computing

In create_matatrix.ipynb, set the mesh size and degree of finite element method.

Here is a sample setting. It defines an uniform mesh with subdivision number 16 for the unit square domain. The FEM is selected as the linear conforming one.

N = 16
mesh = UnitSquareMesh(N,N) 
degree = 1