Steady and unsteady Poisson and Stokes equations are solved using mesh dependent Finite Element Method and meshless Radial Basis Function Collocation Method to compare the performances of these two numerical techniques across several criteria. The accuracy of Radial Basis Function Collocation Method with multiquadrics is enhanced by implementing a shape parameter optimization algorithm. For the time-dependent problems, time discretization is conducted using Backward Euler Method. The performances are assessed over the accuracy, runtime, condition number, and ease of implementation criteria. Three kinds of errors were calculated; least square error, root mean square error and maximum relative error. To calculate the least square error while using meshless Radial Basis Function Collocation Method, a novel technique is implemented. Imaginary numerical solution surfaces are created and then the volume between those imaginary surfaces and the analytic solution surfaces is calculated, enabling a fair error calculation. Lastly, all solutions are put together and solution trends are observed over the number of solution nodes vs. runtime, accuracy vs. runtime, and accuracy vs. the number of nodes. The assessment indicates the criteria under which Finite Element Method perform better and those when Radial Basis Function Collocation Method outperforms its mesh dependent counterpart.
翻译：稳定且不稳定的 Poisson 和 Stokes 等方程式通过网状依赖的有限元素法和网状平面基本功能配置法来解决,以比较这两种数字技术的性能,这些技术可以跨越若干标准。通过实施形状参数优化算法,半径基功能配置法的准确性得到加强。对于时间依赖问题,使用向后倾斜的 Euler 方法进行时间分解。最后,根据精确度、运行时间、条件编号和执行方便度标准来评估性能。计算出三种错误;最小的方形错误、根正方形平均错误和最大相对错误。在使用无光的光基功能定位方法的同时计算最小的方形错误,采用了一种新技术。想象性数字解决方案表面的创建,然后计算出这些假形表面与分析性解决方案表面之间的体积,从而进行公平的误差计算。最后,所有解决方案都组合在一起,在解决方案与运行时间、准确度、运行时间和准确度之间观察到解决方案的趋势。在使用无边框边框边框的公式下,将执行更好的平面方法。
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