Amongst the more exciting phenomena in the field of nonlinear partial
differential equations is the Lavrentiev phenomenon which occurs in the calculus of
variations. We prove that a conforming finite element method fails if and only if the
Lavrentiev phenomenon is present. Consequently, nonstandard finite element methods
have to be designed for the detection of the Lavrentiev phenomenon in the computational
calculus of variations.
We formulate and analyze a general strategy for solving variational problems in the
presence of the Lavrentiev phenomenon based on a splitting and penalization strategy.
We establish convergence results under mild conditions on the stored energy function.
Moreover, we present practical strategies for the solution of the discretized problems
and for the choice of the penalty parameter.
CMAM considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. The journal is interdisciplinary while retaining the common thread of numerical analysis, readily readable and meant for a wide circle of researchers in applied mathematics.