Spurious solutions for high-order curl problems

Abstract

Abstract We investigate numerical solutions of high-order $\operatorname {curl}$ 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 $\operatorname {curl}$ problems by relating the partial differential equations to the Hodge–Laplacian boundary problems of the $\operatorname {grad}\operatorname {curl}$ complexes.