A high-order meshless Galerkin method for semilinear parabolic equations on spheres
A high-order meshless Galerkin method for semilinear parabolic equations on spheres
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DOI：
10.1007/s00211-018-01021-7
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发表时间：
2019-01
影响因子：
2.1
通讯作者：
Jens Künemund;F. Narcowich;J. Ward;H. Wendland
中科院分区：
文献类型：
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作者：
Jens Künemund;F. Narcowich;J. Ward;H. Wendland
本文提出了一种新的无网格Galerkin方法来数值求解球面上的半线性抛物型方程。新的近似方法是基于在空间中使用球面基函数的Galerkin近似的离散化。由于我们的空间近似空间是用球面基函数构建的，因此它们可以是任意阶的，并且不需要构建底层网格。我们将建立无网格方法的收敛性，通过适应，球，收敛结果，由于Zerée和Wahlbin。要做到这一点，需要证明新的近似结果，包括一个新的逆或球形基函数的Nikolskii不等式。我们还讨论了如何在Galerkin方法中的积分可以准确和更有效地计算使用最近开发的求积规则。这些新的求积公式也适用于球面上椭圆型偏微分方程的Galerkin近似。最后，我们提供了几个数值例子。
We describe a novel meshless Galerkin method for numerically solving semilinear parabolic equations on spheres. The new approximation method is based upon a discretization in space using spherical basis functions in a Galerkin approximation. As our spatial approximation spaces are built with spherical basis functions, they can be of arbitrary order and do not require the construction of an underlying mesh. We will establish convergence of the meshless method by adapting, to the sphere, a convergence result due to Thomée and Wahlbin. To do this requires proving new approximation results, including a novel inverse or Nikolskii inequality for spherical basis functions. We also discuss how the integrals in the Galerkin method can accurately and more efficiently be computed using a recently developed quadrature rule. These new quadrature formulas also apply to Galerkin approximations of elliptic partial differential equations on the sphere. Finally, we provide several numerical examples.