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Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010),281-354] that linear, mixed variational problems, and their numericalapproximation by mixed finite element methods, can be studied using thepowerful, abstract language of Hilbert complexes. In another recent article[arXiv:1005.4455], we extended the Arnold-Falk-Winther framework by analyzingvariational crimes (a la Strang) on Hilbert complexes. In particular, this gavea treatment of finite element exterior calculus on manifolds, generalizingtechniques from surface finite element methods and recovering earlier a prioriestimates for the Laplace-Beltrami operator on 2- and 3-surfaces, due to Dziuk[Lecture Notes in Math., vol. 1357 (1988), 142-155] and later Demlow [SIAM J.Numer. Anal., 47 (2009), 805-827], as special cases. In the present article, weextend the Hilbert complex framework in a second distinct direction: to thestudy of semilinear mixed problems. We do this, first, by introducing anoperator-theoretic reformulation of the linear mixed problem, so that thesemilinear problem can be expressed as an abstract Hammerstein equation. Thisallows us to obtain, for semilinear problems, a priori solution estimates anderror estimates that reduce to the Arnold-Falk-Winther results in the linearcase. We also consider the impact of variational crimes, extending the resultsof our previous article to these semilinear problems. As an immediateapplication, this new framework allows for mixed finite element methods to beapplied to semilinear problems on surfaces.

Semilinear mixed problems on Hilbert complexes and their numerical approximation