Abstract
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.
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Notes
On \({\mathbb {S}}^2\), with \(\theta \) being the colatitude and \(\varphi \) being the longitude, the metric ds has the form \(ds^2 = d\theta ^2+ \sin ^2 \theta d\varphi ^2\). Thus the four entries of the tensor are \(g_{11}=1\), \(g_{12}=g_{21}=0\), and \(g_{22}=\sin \theta \).
On \({\mathbb {S}}^2\), in colatitude–longitude coordinates, \(\Delta _* u= \frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\big (\sin \theta \frac{\partial u }{\partial \theta }\big ) + \frac{1}{\sin ^2 \theta }\frac{\partial ^2 u}{\partial \varphi ^2 }\).
One can establish this by showing that the asymptotic estimate in [24, Eq. 4.11], where \(s=m\), is equal to the right-hand side in that inequality.
For d odd, it is an open question as to whether the exponential inequalities hold. There are however bounds in terms of powers of h. See [16, Theorem 5.5].
This choice of \(r_0\) is large, in the sense that the ball \(B(\xi ,r_0)\) has to contain more Y points than the usual \(Kh_Y |\log (h_Y)|\). It should be possible to do better.
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F.J. Narcowich and J.D. Ward: Research supported by Grant DMS-1514789 from the National Science Foundation.
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Künemund, J., Narcowich, F.J., Ward, J.D. et al. A high-order meshless Galerkin method for semilinear parabolic equations on spheres. Numer. Math. 142, 383–419 (2019). https://doi.org/10.1007/s00211-018-01021-7
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DOI: https://doi.org/10.1007/s00211-018-01021-7