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.
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.