In this paper, we design a new framework of direct radial basis function partition of unity (D-RBF-PU) method to solve parabolic equation on surface with and without boundary. Resort to the tangent plane approach, the proposed method avoids dealing with complex surface differentiation operators, and only needs to approximate the standard differentiation operators on the two-dimensional tangent space compared with the existing RBF-PU methods on surface. The new meshfree method is called the “D-RBF-PU tangent plane” method. Additionally, for the surface parabolic equation with Neumann boundary condition, the ghost node technique is considered to avoid the concentration of test nodes on one side of the patch near the boundary and improve the computational accuracy. Numerical examples are performed to confirm the convergence and the eigenvalue stability and applications to the Fitzhugh–Nagumo model and Schnakenberg model on different surfaces demonstrate the efficiency of the proposed method.

中文翻译：

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直接RBF-PU法结合切面法求解曲面抛物线方程


在本文中，我们设计了一种新的直接径向基函数统一划分（D-RBF-PU）方法框架来求解有边界和无边界表面上的抛物线方程。与现有的表面RBF-PU方法相比，该方法借助切平面方法，避免了处理复杂的表面微分算子，只需要在二维切空间上逼近标准微分算子。新的无网格方法称为“D-RBF-PU切平面”方法。另外，对于具有诺伊曼边界条件的曲面抛物型方程，考虑了鬼节点技术，避免测试节点集中在贴片靠近边界的一侧，提高计算精度。数值例子验证了收敛性和特征值稳定性，并且在不同表面上对 Fitzhugh-Nagumo 模型和 Schnakenberg 模型的应用证明了该方法的效率。