For a continuous coercive bilinear form with continuity constant and coercivity constant , a conforming finite-element solution satisfiesFor a symmetric problem measured in its energy norm, the constant is one.
For any , Galerkin orthogonality gives , where . Coercivity and continuity yield . Divide by when it is nonzero and take the infimum. In the symmetric case, the Cauchy-Schwarz inequality in the energy norm gives the constant one.
For a symmetric elliptic variational problem with Galerkin orthogonality, let and solve the dual problem for every . If elliptic regularity gives , then a finite element interpolation estimate givesThus . Combining this with the Céa lemma and an energy norm error gives an L2 norm error. The additional order depends on the dual regularity assumption, which can fail on unsuitable domains.
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Céa's lemma is a result in the field of functional analysis and calculus of variations, particularly in the context of optimal control problems. The lemma is often used to derive estimates for the behavior of solutions to variational problems. In a general sense, Céa's lemma states that under certain conditions, the error in the approximation of a functional can be controlled in terms of the norm of a corresponding linear functional applied to the error of the function.