We investigate numerical solutions of high order curl problems with various formulations and finite elements. We show that several classical conforming finite elements lead to spurious solutions, while mixed formulations with finite elements in complexes solve the problems correctly. To explain the numerical results, we clarify the cohomological structures in high order curl problems by relating the partial differential equations to the Hodge-Laplacian boundary problems of the gradcurl-complexes.

翻译：我们用各种配方和有限元素来调查高序曲线问题的数字解决方案。 我们发现,一些古典的符合有限元素导致虚假解决方案,而复杂元素中含有有限元素的混合配方则正确地解决问题。 为了解释数字结果,我们通过将部分差异方程式与梯子-复合体的霍奇-拉普拉西亚边界问题联系起来,澄清高序曲线的共生结构问题。