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Summary: SPLINE ELEMENT METHOD FOR THE MONGE-AMP`ERE
EQUATION
GERARD AWANOU
Abstract. We consider numerical approximations of the Monge-Amp`ere equation
det D2
u = f, f > 0 with Dirichlet boundary conditions on a convex bounded
domain in Rn
, n = 2, 3. We make a comparative study of three existing methods
suitable for finite element computations. We construct conforming approximations
in the framework of the spline element method where constraints and interelement
continuities are enforced using Lagrange multipliers.
1. Introduction
This paper addresses the numerical solution of the Dirichlet problem for the Monge-
Amp`ere equation
(1.1) det D2
u = f in , u = g on ,
where D2
u = 2u
xixj
i,j=1,...,n
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