Nonvariational

Nonvariational is an adjective used in mathematics and related fields to describe problems, equations, or methods that do not arise from or cannot be expressed as the stationary points of a functional, i.e., they do not have a variational formulation as Euler–Lagrange equations of a functional.

In the context of partial differential equations, a problem is variational if its solutions can be characterized as minimizers or extrema of an energy or action functional. Nonvariational PDEs lack such a reformulation or do not admit a meaningful functional whose critical points correspond to solutions. This distinction has implications for both theory and computation: variational problems allow energy methods, existence proofs via direct methods, and natural numerical schemes based on Galerkin-type weak forms. Nonvariational problems, by contrast, may require alternative approaches that do not rely on an energy principle, such as operator-theoretic techniques, residual-based formulations, or discretizations that work with the strong form in a weak sense.

In numerical analysis, the term nonvariational finite element method (NVFEM) is used to describe discretizations of PDEs that do not possess a standard variational form. NVFEM and related methods aim to provide stable and accurate solutions for nonvariational operators, including non-self-adjoint or non-potential systems, by adapting the finite element framework to accommodate forms that cannot be derived from a functional.

Nonvariational problems commonly appear in contexts involving dissipation, external forcing, or transport-dominated processes, where a traditional energy-based formulation is unavailable.