Hadamard’s Well-Posedness Criteria

In 1902, the mathematician Jacques Hadamard formulated that a boundary value problem (often for partial differential equations, PDEs) is well-posed if it satisfies three properties:

1. Existence of a solution.
2. Uniqueness of that solution.
3. Continuous dependence of the solution on the input data (i.e., small changes in boundary conditions, initial conditions, or parameters lead to small changes in the solution).

These criteria ensure that the problem is both physically and mathematically meaningful:

• If a solution does not exist or is not unique, then the model fails to describe a well-defined physical reality.
• If the solution is extremely sensitive to small perturbations, then small uncertainties in measurements or modeling can cause drastically different outcomes, which is undesirable for engineering analyses.

Well-Posedness in the Context of FEM

Connection to the Underlying PDE

The Finite Element Method is a numerical technique to approximate solutions of PDEs (or integral equations). Whether or not an FEM solution behaves well is intrinsically tied to the well-posedness of the underlying continuous problem.

1. If the PDE is well-posed:
   • The Galerkin or FEM approximation will typically converge to the unique solution as the mesh is refined (under appropriate conditions like stability, coercivity, etc.).
   • Solutions will be relatively robust to small changes in mesh size, boundary conditions, or material parameters.
2. If the PDE is ill-posed:
   • No matter how refined the mesh is, the numerical solution may not converge or may exhibit pathological sensitivity (e.g., localized solutions that shrink to zero width, spurious oscillations, or multiple “solutions”).
   • In such cases, standard FEM will not reliably capture the true physics unless additional regularization or remedial modeling is introduced.

FEM Formulation and Convergence

In simpler, linear PDE problems (e.g., linear elasticity, heat conduction without phase change), well-posedness is often guaranteed by classical theorems such as the Lax–Milgram theorem. This theorem states that if a PDE problem is:

• coercive (the bilinear form is strictly positive),
• bounded (the bilinear form satisfies certain norm bounds), and
• consistent (the load/forcing term belongs to an appropriate dual space),

then the FEM solution exists, is unique, and converges to the real solution as the mesh is refined. This is a direct application of Hadamard’s idea in a functional analysis framework.