On the numerical solution of a two-dimensional Pucci’s On the numerical solution of a two-dimensional Pucci’s equation with Dirichlet boundry conditions: a least-squares approach

Roland Glowinski
University of Houston

In this presentation we will discuss the numerical solution of a two-dimensional, fully nonlinear elliptic equation of the Pucci’s type, completed by Dirichlet boundary conditions, namely:
(PUC-D) a? + + ? – = 0 in O, ? = g on G.

In (PUC-D): (i) O is a bounded domain of R2 with G its boundary. (ii) ? + and ? – are, respectively, the largest and smallest eigenvalues of the Hessian D2? of the unknown function ?. (iii) a ? (1, + ?).
The solution method relies on a least-squares formulation taking place in a subset of H2(O)?Q, where Q is the space of the 2?2 symmetric tensor-valued functions with components in L2(O). After an appropriate space discretization the resulting finite dimensional problem is solved by an iterative method operating alternatively in the spaces Vh and Qh approximating H2(O) and Q, respectively. The results of numerical experiments are presented; they validate the methodology discussed in this lecture.
Additional computational results will be presented; they concern the homogeneization properties of the solutions of the above Pucci’s problem when the coefficient a in the first equation oscillates periodically between two values = 1.

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