Finite-element PDE solver admissibility and certificate contract

Fix an integer $d\ge 2$, a bounded polyhedral open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$, a finite matching affine simplicial mesh ${\mathcal{T}}_h$ of $\mathrm{\Omega }$, and unisolvent reference finite elements assembled on that mesh. Thus the mesh is a finite face-to-face cover by nondegenerate affine simplices with disjoint interiors, and each local finite-element degree-of-freedom map is bijective on its finite-dimensional shape-function space. Fix coefficient spaces ${\mathrm{\Theta }}_h={\mathbb{R}}^m$, $V_h={\mathbb{R}}^n$, and $Y_h={\mathbb{R}}^s$ with their standard Euclidean inner products, where $m,n,s\ge 1$. Let $Q_h\subseteq {\mathrm{\Theta }}_h$ be nonempty, closed, and convex, and let $O_h\subseteq {\mathrm{\Theta }}_h$ be open with $Q_h\subseteq O_h$.

A finite-element PDE model in this contract consists of a continuously differentiable residual map $R_h:O_h\times V_h\to V_h$ and a continuously differentiable observation map $G_h:O_h\times V_h\to Y_h$. It must supply a continuously differentiable state branch $S_h:O_h\to V_h$ satisfying $R_h(q,S_h(q))=0$ for every $q\in O_h$. For every $q\in Q_h$, this branch must be the specified unique admissible state and the matrix $D_u{R}_h(q,S_h(q))$ must be invertible. The discrete forward map is $F_h(q):=G_h(q,S_h(q))$. The unisolvent finite-element degrees of freedom identify $V_h$ with its coefficient space; all matrix transposes below are in the fixed Euclidean coefficient inner products.

For data $y^{\delta }\in Y_h$, reference parameter $q_0\in {\mathrm{\Theta }}_h$, and $\alpha >0$, the discrete nonlinear Tikhonov objective is

Define the finite-dimensional objective before eliminating the state by

The symbols ${\mathrm{\nabla }}_u{J}_h$ and ${\mathrm{\nabla }}_q{J}_h$ denote the Euclidean gradient vectors representing the two partial derivative functionals. At an admissible $q$, the exact discrete adjoint $p_h(q)\in V_h$ is the unique solution of

The associated discrete-adjoint gradient vector is

The next lemma proves that this vector is the gradient of ${\mathrm{\Phi }}_h$; the definition itself asserts no minimizer exists and no gradient iteration converges.

An admissibility predicate is a specified finite-time test ${\mathrm{adm}}_h(q)$ that decides whether $q\in Q_h$ and checks the model's stated coefficient, boundary, mesh, and time-step hypotheses before a PDE solve. For an explicit time discretization, a Courant–Friedrichs–Lewy condition (CFL condition) is a method-specific bound relating the time step to the mesh and the discrete evolution operator; no universal CFL threshold is part of this definition. A true predicate result certifies that the declared model hypotheses hold for that input, not merely that a heuristic check passed.

The model must also supply an exact finite-dimensional projection routine which returns $P_{Q_h}(z)$ for a queried $z\in {\mathrm{\Theta }}_h$ in finite time or reports failure only when its declared finite call budget is exhausted. The unique nearest point exists because $Q_h$ is nonempty, closed, and convex, but an operational routine for computing it is an additional part of this contract.

A solver routine with declared nonnegative state and adjoint tolerances ${\tau }_u,{\tau }_p$ and a finite solve budget must return either failure or approximate state and adjoint vectors $\tilde{u},\tilde{p}$ together with certified finite bounds ${\rho }_u,{\rho }_p,{\eta }_u,{\eta }_p,P,L_{J,u},L_{R,u},M_q,L_{G,u}$. The residual bounds have the exact-system meaning $\mathrm{\parallel }{R}_h(q,\tilde{u}){\mathrm{\parallel }}_{V_h}\le {\rho }_u$ and $\mathrm{\parallel }(D_u{R}_h(q,u){)}^T\tilde{p}-{\mathrm{\nabla }}_u{J}_h(q,u){\mathrm{\parallel }}_{V_h}\le {\rho }_p$, where $u=S_h(q)$; both must satisfy ${\rho }_u\le {\tau }_u$ and ${\rho }_p\le {\tau }_p$. In particular, an adjoint residual evaluated at $\tilde{u}$ alone does not certify ${\rho }_p$ unless the model also bounds the resulting state and derivative-data defects. With $u=S_h(q)$ and $p=p_h(q)$, the model must certify

and $\mathrm{\parallel }{G}_h(q,u)-G_h(q,\tilde{u}){\mathrm{\parallel }}_{Y_h}\le L_{G,u}{\eta }_u$. It then returns

All constants and state/adjoint error bounds must be justified for the chosen model with every derivative-data, geometry, quadrature, and algebraic-solve defect retained. Certified residual bounds ${\rho }_u,{\rho }_p$ do not by themselves imply ${\eta }_u,{\eta }_p$; a separate model-valid stability estimate is required. The proposition below derives the resulting ${\epsilon }_g$ and ${\epsilon }_F$ bounds. An objective-evaluation routine must return in its finite budget either failure or an interval $[{\mathrm{\ell }}_h(q),u_h(q)]$ containing the exact value ${\mathrm{\Phi }}_h(q)$; failure is permitted only when a required state/adjoint solve fails its contract or the evaluation-call budget is exhausted. Exact evaluation is the degenerate-interval case. Projection and objective-evaluation calls consume the algorithm's finite evaluation budget. Each routine must return within its declared budget or report failure.

If a continuous model $F$ is also supplied, an optional mesh certificate ${\epsilon }_{\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{h}}$ must separately bound the claimed continuous-to-discrete error, such as $\mathrm{\parallel }{F}_h(q)-F(q)\mathrm{\parallel }$ or $\mathrm{\parallel }\mathrm{\nabla }{\mathrm{\Phi }}_h(q)-\mathrm{\nabla }\mathrm{\Phi }(q)\mathrm{\parallel }$. The state and adjoint algebraic residual certificates do not imply this mesh certificate.

Nonlinear forward maps on an open real Hilbert-space domain

Nonlinear Tikhonov functional and its domain

Matching affine simplicial meshes and shape regularity

Reference finite elements and unisolvent degrees of freedom

Standard Euclidean inner product and coefficient norm

Existence and uniqueness of Euclidean nearest-point projections

Finite-dimensional Riesz representation

Linear finite-element residual-to-state and adjoint error estimate

Exact discrete adjoint on a finite state-step graph

Continuous elliptic PDE adjoint gradient

Finite-element state and adjoint error bounds for a linear elliptic control model

Hannes Meinlschmidt, Optimization with Partial Differential Equations, complete lecture notes

Michael Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case

Fernando Gaspoz, Christian Kreuzer, Andreas Veeser, and Winnifried Wollner, A Posteriori Error Analysis for Optimization with PDE Constraints

The algorithmic wrapper below is finite-dimensional and uses only supplied finite returns. No axiom of choice is used by its projection or finite-dimensional Riesz steps. A model-specific elliptic state/adjoint theorem may require Countable Choice, and that branch must retain its own hypotheses. The nonlinear tangential-cone and Tikhonov existence/stability results are not convergence theorems for this finite-budget procedure.