We consider finite element approximations for positive solutions of the semilinear elliptic problem $$ -\Delta u=a(x)u^p, $$ with the function $a(\cdot)$ changing sign, and with superlinear growth ($p>1$). We expect solutions of this problem to be saddle-points of the associated energy functional, and therefore minimization techniques are not suitable for this case. However, since solutions may be characterized as constrained maxima for a different functional, we will give a discrete version of this approach and study the convergence of approximate solutions. We also present some numerical experiments.
Ferretti, R., FINZI VITA, S. (1998). On a variational approximation of superlinear indefinite elliptic problems. NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, 19, 759-772.
On a variational approximation of superlinear indefinite elliptic problems
FERRETTI, Roberto;
1998-01-01
Abstract
We consider finite element approximations for positive solutions of the semilinear elliptic problem $$ -\Delta u=a(x)u^p, $$ with the function $a(\cdot)$ changing sign, and with superlinear growth ($p>1$). We expect solutions of this problem to be saddle-points of the associated energy functional, and therefore minimization techniques are not suitable for this case. However, since solutions may be characterized as constrained maxima for a different functional, we will give a discrete version of this approach and study the convergence of approximate solutions. We also present some numerical experiments.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
