An L²-finite element approximation of the solution to div/curl-systems with nonempty null space

Partial differential equations involving divergence or curl operators often introduce nonempty null-space components in their solutions, causing conventional numerical methods to produce inaccurate results. In this paper, we present a new algorithm that addresses this challenge by explicitly accounting for the null-space component. The dual system least squares method is used to identify this component and remove it from the approximation using negative-norm minimization. Our approach retains the use of H¹-conforming basis functions, ensuring that the resulting approximation is orthogonal to the null space of the operator. The effectiveness of our algorithm is demonstrated through numerical experiments with various partial differential equations, showing improved accuracy compared to conventional methods.