A Finite Difference Method for the Variational p-Laplacian

We propose a new monotone finite difference discretization for the variational p -Laplace operator, Δ p u = div ( | ∇ u | p - 2 ∇ u ) , and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-...

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Bibliographic Details
Published in:Journal of scientific computing 2022, Vol.90 (1), p.67, Article 67
Main Authors: del Teso, Félix, Lindgren, Erik
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Computational Mathematics and Numerical Analysis
Dirichlet problem
Discretization
Dynamic programming principle
Finite difference
Finite difference method
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Mean value property
Methods
Nonhomogeneous
Nonlinear systems
p-Laplacian
Theoretical
Viscosity
Viscosity solutions
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Summary:We propose a new monotone finite difference discretization for the variational p -Laplace operator, Δ p u = div ( | ∇ u | p - 2 ∇ u ) , and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p -Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.
ISSN:0885-7474
1573-7691
1573-7691
DOI:10.1007/s10915-021-01745-z