## Computing the first eigenpair of the $$p$$-Laplacian via inverse iteration of sublinear supersolutions.(English)Zbl 1255.65205

Summary: We introduce an iterative method for computing the first eigenpair $$(\lambda _{p },e _{p })$$ for the $$p$$-Laplacian operator with homogeneous Dirichlet data as the limit of $$(\mu _{q,} u _{q })$$ as $$q\rightarrow p ^{ - }$$, where $$u _{q }$$ is the positive solution of the sublinear Lane-Emden equation $$-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$$ with the same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of $$u _{q }$$ to $$e _{p }$$ is in the $$C ^{1}$$-norm and the rate of convergence of $$\mu _{q }$$ to $$\lambda _{p }$$ is at least $$O(p - q)$$. Numerical evidence is presented.

### MSC:

 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs

CUBIT; PETSc
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