2011/03/09 by Susanne Brenner, Susanne C. Brenner, Thirupathi Gudi +2 · 9 citations
Mathematics · #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.1090/s0025-5718-2011-02487-7
In this paper, we develop and analyze <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Superscript 0"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal C0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> penalty methods for the fully nonlinear Monge-Ampère equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="det left-parenthesis upper D squared u right-parenthesis equals f"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">det</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">det (D2 u) = f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.