A HYBRID ANALYTICAL-NUMERICAL TECHNIQUE FOR ELLIPTIC PDES
A HYBRID ANALYTICAL-NUMERICAL TECHNIQUE FOR ELLIPTIC PDES
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DOI：
10.1137/18m1217309
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发表时间：
2019-01-01
影响因子：
3.1
通讯作者：
Hashemzadeh, Parham
中科院分区：
文献类型：
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作者：
Colbrook, Matthew J.;Fokas, Thanasis S.;Hashemzadeh, Parham
最近的工作已经产生了一种新的和简单的数值方法来解决椭圆边值问题制定在凸多边形的二维。该方法，基于统一的变换，涉及扩大未知的边界值在勒让德基础和确定的膨胀系数，通过评估所谓的全球关系在适当的点在复杂的傅立叶平面（频谱配置）。在本文中，我们提供了一个显着的进步，这种数值技术提供了一个快速，有效的方法来评估的解决方案在域内部。勒让德基的使用允许使用切比雪夫插值高效且准确地计算相关积分，即使是在很大程度上。对于特殊情况下的拉普拉斯方程，这允许一个明确的扩展在域内部的超几何函数。在内部的评估被发现收敛速度更快，比未知的边界值的近似，允许精确近似的解决方案与弱角奇异性。对于较强的奇异性，该方法可以与全局奇异函数相结合，以快速收敛。数值例子表明，该方法与标准谱方法相比，开辟了将该方法应用于一般曲线域的可能性。
Recent work has given rise to a novel and simple numerical technique for solving elliptic boundary value problems formulated in convex polygons in two dimensions. The method, based on the unified transform, involves expanding the unknown boundary values in a Legendre basis and determining the expansion coefficients by evaluating the so-called global relation at appropriate points in the complex Fourier plane (spectral collocation). In this paper we provide a significant advancement of this numerical technique by providing a fast and efficient method to evaluate the solution in the domain interior. The use of a Legendre basis allows the relevant integrals to be computed efficiently and accurately using Chebyshev interpolation, even for large degree. For the particular case of the Laplace equation this allows an explicit expansion in the domain interior in terms of hypergeometric functions. Evaluation in the interior is found to converge more rapidly than the approximation of the unknown boundary values, allowing accurate approximation of solutions with weak corner singularities. For stronger singularities, the method can be combined with global singular functions for rapid convergence. Numerical examples are provided, showing that the method compares well against standard spectral methods and opens up the possibility of applying the method to general curvilinear domains.