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Hermitian Galerkin solutions are best in the energy norm

Statement

Assume the hypotheses and notation of Céa quasi-optimality for coercive conforming Galerkin problems, and suppose additionally that $a$ is Hermitian: $a (v, w) = \overline{a (w, v)}$ . Then $∥ v ∥_{a} : = \sqrt{a (v, v)}$ is the energy norm and the Galerkin solution is its best approximation: $∥ u - u_{h} ∥_{a} = \inf_{v_{h} \in V_{h}} ∥ u - v_{h} ∥_{a} .$

Facts & Assumptions

Given: The bounded coercive Galerkin setting of the cited Céa theorem and the additional Hermitian identity for $a$ .

[F1]

The Céa theorem supplies unique continuous and discrete solutions (Céa quasi-optimality for coercive conforming Galerkin problems).

[F2]

The error $u - u_{h}$ is orthogonal to $V_{h}$ in the form $a$ (Galerkin orthogonality for conforming approximations).

[F3]

A bounded coercive form is Hermitian exactly when $a (v, w) = \overline{a (w, v)}$ ; coercivity gives $Re a (v, v) \geq α ∥ v ∥_{V}^{2}$ (Bounded, coercive and symmetric sesquilinear forms).

Proof

1.1F3given

Hermitian symmetry makes $a (v, v)$ real, and coercivity makes it positive for every nonzero $v$ . Thus $a$ is an inner product and $∥ \cdot ∥_{a}$ is its induced norm.

1.2F2F3given

Put $e = u - u_{h}$ and, for arbitrary $v_{h} \in V_{h}$ , put $η_{h} = u_{h} - v_{h}$ . By [F2], $a (e, η_{h}) = 0$ ; Hermitian symmetry gives $a (η_{h}, e) = \overline{a (e, η_{h})} = 0$ .

2.1F2F3step 1.2algebra

Since $u - v_{h} = e + η_{h}$ , sesquilinearity and step 1.2 give $∥ u - v_{h} ∥_{a}^{2} = a (e + η_{h}, e + η_{h}) = ∥ e ∥_{a}^{2} + ∥ η_{h} ∥_{a}^{2} \geq ∥ e ∥_{a}^{2} .$ Therefore $∥ u - u_{h} ∥_{a} \leq ∥ u - v_{h} ∥_{a}$ for every $v_{h} \in V_{h}$ .

3.1F1step 2.1algebra∎

Taking the infimum in step 2.1 gives one inequality. The choice $v_{h} = u_{h}$ gives the reverse inequality, so equality holds.

Depends on

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part VI, Chapter 26, Basic error analysis (standard reference, not scraped)
Thomas Richter, Die Finite Elemente Methode für partielle Differentialgleichungen (standard reference, not scraped)