Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation |
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Authors: | Michael Holst Ari Stern |
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Institution: | 1.Department of Mathematics,University of California, San Diego,La Jolla,USA |
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Abstract: | Arnold, Falk, and Winther recently showed (Bull. Am. Math. Soc. 47:281–354, 2010) that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied
using the powerful, abstract language of Hilbert complexes. In another recent article (arXiv:), we extended the Arnold–Falk–Winther framework by analyzing variational crimes (à la Strang) on Hilbert complexes. In particular,
this gave a treatment of finite element exterior calculus on manifolds, generalizing techniques from surface finite element
methods and recovering earlier a priori estimates for the Laplace–Beltrami operator on 2- and 3-surfaces, due to Dziuk (Lecture Notes in Math., vol. 1357:142–155,
1988) and later Demlow (SIAM J. Numer. Anal. 47:805–827, 2009), as special cases. In the present article, we extend the Hilbert complex framework in a second distinct direction: to the
study of semilinear mixed problems. We do this, first, by introducing an operator-theoretic reformulation of the linear mixed
problem, so that the semilinear problem can be expressed as an abstract Hammerstein equation. This allows us to obtain, for
semilinear problems, a priori solution estimates and error estimates that reduce to the Arnold–Falk–Winther results in the linear case. We also consider
the impact of variational crimes, extending the results of our previous article to these semilinear problems. As an immediate
application, this new framework allows for mixed finite element methods to be applied to semilinear problems on surfaces. |
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