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Patra, Suchismita; Srinivas Kumar, V. V. K.

A local min-orthogonal based numerical method for computing multiple coexisting solutions to cooperative \(p\)-Laplacian systems. (English) Zbl 1479.65031

Appl. Numer. Math. 169, 221-242 (2021).
Summary: The objective of the present paper is to provide a detailed account of a local min-orthogonal method for finding multiple coexisting solutions to cooperative \(p\)-Laplacian systems. A local \(L\)-\(\bot\) selection mapping on a product space of two Banach spaces is introduced in order to develop a local characterization of the coexisting solutions. Using this local characterization we propose a numerical algorithm for finding multiple coexisting solutions to the system and provide a subsequence convergence result of the algorithm. We also discuss the discretization of the problem using the finite element method and the associated convergence results of the numerical solutions. Finally, we provide the numerical results to demonstrate the efficiency of the proposed method and also to confirm its related theoretical results.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N99 Numerical methods for partial differential equations, boundary value problems
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35D30 Weak solutions to PDEs
49J35 Existence of solutions for minimax problems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

cooperative elliptic systems; \(p\)-Laplacian systems; multiple critical points; local min-orthogonal method; finite element method
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References:

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