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Amongst the more exciting phenomena in the field of nonlinear partialdifferential equations is the Lavrentiev phenomenon which occurs in the calculus ofvariations. We prove that a conforming finite element method fails if and only if theLavrentiev phenomenon is present. Consequently, nonstandard finite element methodshave to be designed for the detection of the Lavrentiev phenomenon in the computationalcalculus of variations.We formulate and analyze a general strategy for solving variational problems in thepresence 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 problemsand for the choice of the penalty parameter.

Analysis of a Class of Penalty Methods for Computing Singular Minimizers