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Background:

Lagrange multiplier method has been employed in numerous fields, such as contact problems, material interfaces, phase transformation, stiff constraints or sliding along interfaces.
It is well known that a bad choice or design of Lagrange multiplier space will produce oscillatory results (unstable problem) on Lagrange multipliers. A huge amount of literature has illustrated this observation and some modifications or improvements have been made to remove oscillations which are typically incurred by deviation of inf-sup condition.

Question:

When reading literature on XFEM, I came across the below argument highlighted in red, which is quite mathematical. How to interpret or understand the space is locally too rich and then as a result the inf-sup condition violates? Thanks for any contribution.

Lagrange multipliers

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The saddle point matrix you must solve takes the following form: $$\begin{bmatrix} A & B^T \\ B \end{bmatrix},$$ where $A$ is the unconstrained matrix and $B$ is the matrix that observes the values of the solution at nodes along the immersed interface.

The conditioning of the saddle point matrix is, in part, controlled by the minimum singular value of $B$. The smaller the minimum singular value of $B$, the worse the conditioning. For details I highly recommend the following paper:

Krendl, Wolfgang, Valeria Simoncini, and Walter Zulehner. "Stability estimates and structural spectral properties of saddle point problems." Numerische Mathematik 124.1 (2013): 183-213. https://arxiv.org/pdf/1202.3330.pdf

Now let us look in detail at the matrix $B$ for an interface immersed in a mesh like in the following picture (black is the mesh, red is the immersed interface):

boundary locking

The matrix $B$ takes the following form:

$$B = \begin{bmatrix} 1/2 & 1/2 \\ & 1/2 & 1/2 \\ & & 1/2 & 1/2 \\ &&& \ddots& \ddots \end{bmatrix}$$

But if you right-multiply $B$ by a vector that oscillates highly, the result is small, since the positive and negative contributions tend to cancel out. For example:

$$\underbrace{\begin{bmatrix}1 & -1 & 1 & -1 &\dots\end{bmatrix}}_{\text{large norm}} ~B = \underbrace{\begin{bmatrix}1/2 & 0 & 0 & 0 & \dots\end{bmatrix}}_{\text{small norm}}$$

So, $B$ has small singular values, corresponding to oscillatory interface vectors. Hence the saddle point matrix is ill-conditioned, and it gets worse the finer the mesh is.

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Each Lagrange multiplier corresponds to a constraint. So if the space of Lagrange multipliers is too large, then you have too many constraints that can no longer be all satisfied at the same time without significantly restricting the number of unknowns you have available to satisfy the physics of the problem. This is what happens in locking: each constraint reduces the number of unknowns by one, and you end up with too few unknowns to be physically accurate.

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