A Fourier-Legendre spectral method for approximating the minimizers of 𝜎_{2,𝑝}-energy

Abstract

This paper proposes a Fourier-Legendre spectral method to find the minimizers of a variational problem, called σ 2 , p \sigma _{2,p} -energy, in polar coordinates. Let X ⊂ R n {\mathbb {X}}\subset \mathbb {R}^n be a bounded Lipschitz domain and consider the energy functional ( 1.1 ) (1.1) whose integrand is defined by W ( ∇ u ( x ) ) ≔ ( σ 2 ( u ) ) p 2 + Φ ( det ∇ u ) {\mathbf {W}}(\nabla u(x))≔(\sigma _2(u))^{\frac {p}{2}}+\Phi (\det \nabla u) over an appropriate space of admissible maps, A p ( X ) \mathcal {A}_p({\mathbb {X}}) . Using Fourier and Legendre interpolation errors, we obtain an error estimate for the energy functional and prove a convergence theorem for the proposed method. Furthermore, we apply the gradient descent method to solve a nonlinear algebraic system which is obtained by discretizing the Euler-Lagrange equations. The numerical experiments are performed to demonstrate the accuracy and effectiveness of our method.