A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon
A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon
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DOI：
10.1093/imanum/drn079
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发表时间：
2010-10-01
影响因子：
2.1
通讯作者：
Fokas, A. S.
中科院分区：
文献类型：
--
作者：
Smitheman, S. A.;Spence, E. A.;Fokas, A. S.
利用绿色定理可以得到拉普拉斯方程和修正的亥姆霍兹方程的解的积分表示。然而，这些表示涉及到解及其在边界上的法向导数，并且对于适定边值问题（BVP），这些函数中的一个是未知的。从狄利克雷数据确定诺依曼数据被称为构造狄利克雷到诺依曼映射。在Fokas（1997，Proc. R. Soc. Lond. A，53，1411-1443）用于求解线性和可积非线性偏微分方程（PDE）的BVP。对于线性偏微分方程，这种方法可以被认为是类似的绿色函数的方法在傅立叶平面。在这种方法中，狄利克雷-诺依曼映射的特征在于一个特定的方程，即所谓的全局关系，它在复k平面上形成，其中k表示谱（傅立叶）变量的复延拓。在这里，我们解决的整体关系数值的拉普拉斯和修改的亥姆霍兹方程在凸多边形。这是通过在谱（傅立叶）平面中适当选择的一组点处评估全局关系来实现的，这就是为什么这种方法被称为“谱配置法”。数值实验表明，该方法继承了用于展开未知函数的基的收敛阶，即多项式基（如Chebyshev基）的指数收敛阶和Fourier基的代数收敛阶.然而，多项式基的相关线性系统的条件数比傅立叶基的高得多。
Integral representations for the solutions of the Laplace and modified Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the solution and its normal derivative on the boundary, and for a well-posed boundary-value problem (BVP) one of these functions is unknown. Determining the Neumann data from the Dirichlet data is known as constructing the Dirichlet-to-Neumann map. A new transform method was introduced in Fokas (1997, Proc. R. Soc. Lond. A, 53, 1411-1443) for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs). For linear PDEs this method can be considered as the analogue of the Green's function approach in the Fourier plane. In this method the Dirichlet-to-Neumann map is characterized by a certain equation, the so-called global relation, which is formulated in the complex k-plane, where k denotes the complex extension of the spectral (Fourier) variable. Here we solve the global relation numerically for the Laplace and modified Helmholtz equations in a convex polygon. This is achieved by evaluating the global relation at a properly chosen set of points in the spectral (Fourier) plane, which is why this method has been called a 'spectral collocation method'. Numerical experiments suggest that the method inherits the order of convergence of the basis used to expand the unknown functions, namely, exponential for a polynomial basis such as Chebyshev, and algebraic for a Fourier basis. However, the condition number of the associated linear system is much higher for a polynomial basis than for a Fourier one.