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Numerical approximation of the first eigenpair of the \(p\)-Laplacian using finite elements and the penalty method. (English) Zbl 0884.65103

For \(1<p <\infty\) and for bounded convex domains \(\Omega\), the authors are interested in the first eigenvalue \(\lambda_1\) of \[ \text{div} \bigl(|\nabla |^{p-2} \nabla u\bigr) =\lambda |u|^{p-2} u\quad \text{in } \Omega, \quad u=0 \quad \text{on } \partial \Omega, \] where \(|\cdot |\) denotes the norm in \(L^2(\Omega)\). The eigenvalue \(\lambda_1\) is simple, and the eigenfunction \(u_1\) of one sign. \(\lambda_1\) is the minimum of \(F(u)= \int_\Omega |\nabla u|^p dx\) over \(W_0^{1,p} (\Omega)\) under the constraint \(G(u) =1- |u|^p =0\). To evaluate \(\lambda_1\) and \(u_1\), the sequence of penalty functionals \(L(u,\gamma_k) =F(u) +\gamma_k [G(u)]^2\) is introduced, and \(u_1\) is approximated by finite elements. Convergence is proved, and numerical results are presented for various values of \(p\) and for \(\Omega= (0,1)\), \(\Omega= (0,1) \times(0,1)\), and \(\Omega =\) unit ball in \(\mathbb{R}^2\).

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
35J70 Degenerate elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
49R50 Variational methods for eigenvalues of operators (MSC2000)

Software:

Mathematica
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References:

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