A novel least squares approach generating approximations orthogonal to the null space of the operator

We introduce a novel least squares functional specifically formulated to solve linear partial differential equations with operators that have a nonempty null space. Our method involves projecting the solution onto the orthogonal complement of the operator's null space to overcome challenges encountered by conventional numerical methods when nonzero null components are present. We describe the theoretical framework of the proposed method and validate it through numerical examples that show improved accuracy and usability in cases where traditional methods are less effective due to significant null space components. Overall, this approach provides a practical and reliable solution for partial differential equations with substantial null space components.