Numerical studies of the Steklov eigenvalue problem via conformal mappings

Abstract In this paper, spectral methods based on conformal mappings are proposed to solve the Steklov eigenvalue problem and its related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use a gradient ascent approach to find the optimal domain which maximizes kth Steklov eigenvalue with a fixed area for a given k. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different k.

[1]  H C Mayer,et al.  Walking with coffee: why does it spill? , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Felipe Lepe,et al.  A Virtual Element Method for the Steklov Eigenvalue Problem Allowing Small Edges , 2015, Journal of Scientific Computing.

[3]  R. Ibrahim.  Liquid Sloshing Dynamics: Theory and Applications , 2005 .

[4]  Hai Bi,et al.  A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem , 2011, Appl. Math. Comput..

[5]  Yidu Yang,et al.  A two-grid discretization scheme for the Steklov eigenvalue problem , 2011 .

[6]  C. Kao,et al.  A moving boundary model motivated by electric breakdown: II. Initial value problem , 2009, 0908.2521.

[7]  Braxton Osting,et al.  Computational Methods for Extremal Steklov Problems , 2016, SIAM J. Control. Optim..

[8]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[9]  S. Tanveer,et al.  The stability of a two‐dimensional rising bubble , 1995 .

[10]  Iosif Polterovich,et al.  SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM , 2014, 1411.6567.

[11]  Iosif Polterovich,et al.  Shape optimization for low Neumann and Steklov eigenvalues , 2008, 0811.2617.

[12]  Jimmy Lamboley,et al.  An extremal eigenvalue problem for the Wentzell-Laplace operator , 2014, 1401.7098.

[13]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[14]  Hehu Xie,et al.  A type of multilevel method for the Steklov eigenvalue problem , 2014 .

[15]  Pedro R. S. Antunes,et al.  Convex shape optimization for the least biharmonic Steklov eigenvalue , 2013 .

[16]  Bodo Dittmar,et al.  Sums of reciprocal Stekloff eigenvalues , 2004 .

[17]  Manoj Kumar,et al.  Simulation of a nonlinear Steklov eigenvalue problem using finite-element approximation , 2010 .

[18]  Yidu Yang,et al.  An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem , 2016, 1601.01429.

[19]  Robert Weinstock,et al.  Inequalities for a Classical Eigenvalue Problem , 1954 .

[20]  Beniamin Bogosel,et al.  The method of fundamental solutions applied to boundary eigenvalue problems , 2016, J. Comput. Appl. Math..

[21]  Jan Sokolowski,et al.  Introduction to shape optimization , 1992 .

[22]  Iosif Polterovich,et al.  The legacy of Vladimir Andreevich Steklov , 2014 .

[23]  Steven R. Bell,et al.  The Cauchy Transform, Potential Theory and Conformal Mapping , 2015 .

[24]  Tobin A. Driscoll,et al.  Schwarz-Christoffel Toolbox User''s Guide , 1994 .

[25]  Julián Fernández Bonder,et al.  Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach , 2006, Annali di Matematica Pura ed Applicata.

[26]  Andrey B. Andreev,et al.  Isoparametric finite-element approximation of a Steklov eigenvalue problem , 2004 .

[27]  J. W. Brown,et al.  Complex Variables and Applications , 1985 .

[28]  B. Bogosel,et al.  Optimal Shapes Maximizing the Steklov Eigenvalues , 2017, SIAM J. Math. Anal..

[29]  Thomas K. DeLillo,et al.  The accuracy of numerical conformal mapping methods: a survey of examples and results , 1994 .