A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalue

Jing An,Zhendong Luo,Hai Bi

doi:10.1186/s13660-016-1158-1

Abstract

In this study, a highly efficient spectral-Galerkin method is posed for the fourth-order Steklov equation with boundary eigenvalue. By making use of the spectral theory of compact operators and the error formulas of projective operators, we first obtain the error estimates of approximative eigenvalues and eigenfunctions. Then we build a suitable set of basis functions included in $H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$ and establish the matrix model for the discrete spectral-Galerkin scheme by adopting the tensor product. Finally, we use some numerical experiments to verify the correctness of the theoretical results.

Highlights

Increasing attention has recently been paid to numerical approximations for Steklov equations with boundary eigenvalue, arising in fluid mechanics, electromagnetism, etc
The fourth-order Steklov equations with boundary eigenvalue have been used in both mathematics and physics, for example, the main eigenvalues play a very key role in the positivity-preserving properties for the biharmonic-operator under the border conditions w = w – χ wν = on ∂
Compared with finite element methods, spectral methods have the characteristics of high accuracy

Summary

Introduction

Increasing attention has recently been paid to numerical approximations for Steklov equations with boundary eigenvalue, arising in fluid mechanics, electromagnetism, etc. (see, e.g., [ – ]). We take into account the following fourth-order Steklov equation with boundary eigenvalue:. ( ) We adopt the generalized Jacobian polynomial to deduce in detail the error formula of the high dimensional projective operator associated with the fourth-order Steklov equation with boundary eigenvalue. By employing the spectral method of compact operators, we obtain the satisfactory error formulas of approximative eigenvalues and eigenfunctions. ), we obtain the following equivalent operator formula:. From the spectral method of compact operators (see, e.g., [ , ]), we deduce that all eigenvalues of are real and finite multiple numbers, which are increasingly arranged as follows:. Seek wN ∈ XN that satisfies a(wN , υ) = b(f , υ), ∀υ ∈ XN It follows from the Lax-Milgram theorem that for By combining Theorem . and ( . ) in [ ], we immediately obtain the desired conclusion

Efficient implementation of the spectral-Galerkin solutions

Numerical examples

Conclusions