Institute for Numerical Simulation
Rheinische Friedrich-Wilhelms-Universität Bonn

Finite Element Method for the Stokes problem

We consider the following Stokes equation. Usually, this describes the flow inside Ω where u denotes the velocity and p the pressure.

1. Strengthened Cauchy-Schwarz constant:

In order to be able to dynamically evaluate the error within a triangular element in a mesh, the strengthned Cauchy-Schwarz inequality is used. For a triangle T having angles α, β, and θ=π-α-β,

  • V(T) is the space spanned by the linear Lagrange bases at a1, a2 and a3.
  • Z(T) is the space spanned by the linear Lagrange bases at b1, ..., b7.
Second eigenvalue
We have

where the constant γ is independent of the following entities
  • h(T):=diameter of T
  • ρ(T):=smallest circle containing T
  • R(T):=largest circle contained in T
  • μ(T):=surface area of T
Second eigenvalue

The stiffness matrix with respect to the bases of V(T) and Z(T) is given in M as follows. In function of its block matrices, the original strengthned Cauchy-Schwarz constant is expressed otherwise
Therefore, γ is given by the square root of the largest eigenvalue of the generalized eigenproblem

We denote a:=cot(α), b:=cot(β), c:=cot(γ), ζ:=a+b+c.
The initial generalized eigenvalue problem is reduced to the following 3-by-3 eigenvalue problem

One can explicitly express the two nonzero eigenvalues λ11(x1,x2,x3,x4) and λ22(x1,x2, x3,x4) as rational functions of x1:=cos(α), x2:=cos(β), x3:=sin(α), x4:=sin(β). Next you see the plots of those two nonzero eigenvalues in terms of the angles α and β

First eigenvalue Second eigenvalue

2. Stokes using Crouzeix-Raviart element:
The former Stokes equation corresponds to the following weak formulation

We use here non-conforming discrete approximations. In fact, as approximating spaces, we use the Crouzeix-Raviart elements Vh for the velocity and the piecewise constant space Qh for the pressure. More precisely, if we denote by [vh] the jump at the edges, we have

Since we use non-conforming approximations, we use the following bilinear forms and we consider then the corresponding linear equations with respect to the above spaces Vh and Qh

We aim at using an error estimator to construct an automatic self-adaptive mesh refinement algorithm which starts from a coarse mesh. Basically, a-posteriori error estimators permit to evaluate the finite element errors without knowing the exact solution. That feature makes it possible to dynamically identify regions where one should have further refinements if the error there is too large. Therefore, adaptive refinements are mainly based on the quality of a-posteriori error estimators. Based upon the above strengthned Cauchy-Schwartz inequality, one can deduce an a-posteriori error estimator which is both efficient and reliable. In other words, one can derive the next property

where the constants c1 and c2 are independent of the discretization properties of the underlaying mesh.
3. Adaptive mesh refinement:
As a first example, we consider a problem having a singularity which is located at a reentrant corner. The domain of simulation is

We start from a very coarse initial discretization. Afterwards, we apply the a-posteriori error estimator ηT to find the elements T to be refined. We refine the M elements (say M=10) that admit the largest values of the estimator. Besides, some local mesh enhancement and updating are applied. That is, we apply local Delaunay mesh improvement based on edge flippings. That mesh improvment does not change the local density of the gridnodes. The mesh distribution is unchanged because the coordinates of the internal nodes remain the same as only the shape of the triangles are improved. As for boundary nodes, they are shifted toward the boundary when there are boundary refinements. We display in the next figure the coarsest mesh, the finest mesh together with a magnification of the corner singularity.

Similarly, we performed a simulation by treating a crack problem where the domain of simulation is

We use the same methodology using an initial coarsest mesh. We apply the same a-posteriori error estimator to obtain the final mesh in the next Figure.

Last Update: on March 31st, 2010, in Bonn, by Maharavo Randrianarivony.