Consistent variation occurs in papers multiple times when reading . So what does it really mean from a maths view? I searched online and found an answer for it. Someone explained it as ‘Solution is consistent if when you plug it to the strong form, the solution satisfies it. For example, a penalty method is not consistent; however, a Nitsch method is’. Can anyone shed some light on this please?


1 Answer 1


The variational method is consistent if it converges to exact solution for series of mesh refinements.

For penalty method, $\gamma$ is a model parameter, whereas for Nitsche method $\gamma$ is stabilisation parameter, which controls convergence rate, and for any stable $\gamma$ numerical solution converges to an exact solution.

Suppose that the exact solution of a continuous problem with constraints ($\mathcal{C}\mathbf{u} - \mathbf{g} = \mathbf{0}$), is exactly interpolated on the finite element approximation space. We show that Nitsche's method is consistent with such an exact solution. Taking the weak equation

\begin{equation} \begin{split} a(\mathbf{u},\mathbf{v}) - \sum_E \int_\Gamma \mathbf{t}^{\textrm{T}}(\mathbf{u}) \phi\gamma\mathcal{P}\mathbf{t}(\mathbf{v}) \textrm{d}\Gamma + \sum_E \int_\Gamma \frac{1}{\gamma_0} \left\{ \mathcal{R}(\mathcal{C}\mathbf{u}-\mathbf{g}-\gamma\mathcal{C}\mathbf{t}(\mathbf{u})) \right\}^\textrm{T} \mathcal{R}(\mathcal{C}\mathbf{v}-\phi\gamma\mathcal{C}\mathbf{t}(\mathbf{v})) \textrm{d}\Gamma = 0 \end{split} \end{equation} where $\mathcal{R} = \mathcal{C}^\textrm{T}(\mathcal{C}\mathcal{C}^\textrm{T})^{-1}$, $\mathcal{P}=\mathcal{R}\mathcal{C}$ and $\phi$ takes values $-1,0,1$, respectively for skew-symmetric, non-symmetric, symmetric variant of Nitche method, respectively. Substituting the tractions approximation \begin{equation} \mathbf{t}(\mathbf{u}) = -\frac{1}{\gamma} \mathcal{R}(\mathcal{C}\mathbf{u}-\mathbf{g}-\gamma\mathcal{C}\mathbf{t}(\mathbf{u})), \end{equation} as a result of simple algebraic manipulations, the standard weak from is recovered \begin{equation} a(\mathbf{u},\mathbf{v}) - \int_\Gamma {\boldsymbol\sigma}^\textrm{T}(\mathbf{u})\mathbf{N}^\textrm{T}\mathbf{v} \textrm{d} \Gamma = 0. \end{equation}

This shows the solution of Nitsche's method is a solution of original problem with constraints. In other words, as long as the problem is well-posed, solution of Nitsche's method converges to the original problem with the given constraints. Note that penalty method solution is not consistent, in a sense that the solution of the original problem with constraints is not a solution when penalty terms enforce constraints.

In using the classical penalty method, to verify correctness of the results one needs to show that results converge for both increasing mesh density (approximation order) and for $\gamma_0 \to 0$. Since Nitsche's method is consistent, it is only necessary to demonstrate solution convergence when we refining mesh or increasing approximation order. Here $\gamma_0$ is understood as the stabilisation parameter which controls the convergence rate. However, since we work with finite precision arithmetics, $\gamma_0$ cannot be too small to have well a conditioned matrix.

More information about notation above and more detail explanation you will find here http://mofem.eng.gla.ac.uk/mofem/html/nitsche_periodic.html where Nitche method is used to enforce periodic constraints.


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