The exact formula of the optimal penalty parameter value of the spectral penalty method for differential equations

Abstract

Spectral penalty methods were originally introduced to deal with the stability of the spectral solution coupled with the boundary conditions for differential equations [Funaro 1986, Funaro and Gottlieb 1988]. Later the penalty method was used for spectral methods to be implemented in irregular domains in multiple dimensions. It has been also shown that there are close relations between discontinuous Galerkin methods and spectral penalty methods in multi-domain and element setting. In addition to stability, the penalty method provides a better accuracy because of its asymptotic behavior in the neighborhood of boundaries. The optimal value of the penalty parameter for accuracy has not been studied thoroughly for the exact form. In this short note, we consider a simple differential equation and study the optimal value of the penalty parameter by minimizing the error in maximum norm. We focus on the optimization for the case of Chebyshev spectral collocation method. We provide its exact form and verify it numerically.

Introduction

Spectral methods are still popular in solving differential equations in various application areas. Spectral methods provide highly accurate solutions to differential equations when solutions are smooth. If the solution is smooth enough, the spectral approximation converges exponentially fast. Spectral methods as a global method are known to be superior to local methods such as finite difference methods although for non-smooth problems finite difference approach such as compact scheme may yield more accurate results than spectral method [1]. For its high accuracy spectral methods are also used recently for fractional differential equations [9].

Stability and accuracy of spectral solutions can be improved further by imposing the given boundary conditions weakly, i.e. by penalizing the boundary conditions. Here we note that a slight different approach is also possible such as the one found in Driscoll and Hale [2] instead of removing or weakening the boundary conditions in the spectral formulation. The spectral penalty method was originally proposed in Funaro [3] for solving elliptic problems and later for hyperbolic problems in Funaro and Gottlieb [4]. The flexibility of the penalty method makes the spectral method suitable for dealing with boundary conditions in irregular domains. For this reason, it is also known that there are close relations between discontinuous Galerkin methods and spectral methods with multi-domains and elements. Extensive works on penalty methods have been made (for more discussions see [6]). There have been rigorous studies on the stability of the spectral penalty method [5]. Penalty methods can be also used for fractional differential equations [8]. The study on the optimization of its accuracy rather relies on numerical experiments, e.g. [7]. In this short note, we consider a simple differential equation and derive the optimal penalty parameter value that minimizes the maximum error. We verify that the derived formula matches the expected results well numerically. We note that our exact formula is valid only for ODEs of the type in Eq. (2.1).