Numerical analysis Boundary value problems and finite elements Theorem

Petrov–Galerkin method

Builds on Hilbert space Bilinear form Continuous dual space Galerkin method

Statement

Let $U$ and $V$ be real Hilbert spaces. Let $a \colon U \times V \to \R$ be a bilinear form with $\abs{a(w, v)} \le M \norm{w}_U \norm{v}_V$ for all $w \in U$ and $v \in V$ , and let $f$ be in the continuous dual space $V'$ . Let $u \in U$ satisfy $a(u, v) = f(v)$ for all $v \in V$ . Let $U_h \subseteq U$ (trial space) and $V_h \subseteq V$ (test space) be finite-dimensional subspaces of the same dimension, with $\beta_h = \inf_{w \in U_h \setminus \set{0}} \ \sup_{v \in V_h \setminus \set{0}} \frac{a(w, v)}{\norm{w}_U \norm{v}_V} > 0.$ Then there is a unique $u_h \in U_h$ with $a(u_h, v) = f(v)$ for all $v \in V_h$ , and $\norm{u - u_h}_U \le \Big( 1 + \frac{M}{\beta_h} \Big) \inf_{w \in U_h} \norm{u - w}_U.$

Remarks

In bases of $U_h$ and $V_h$ the discrete problem is a square linear system. The condition $\beta_h > 0$ says that $a(w, v) = 0$ for all $v \in V_h$ forces $w = 0$ , so the system matrix is injective and therefore invertible. Subtracting the equations gives $a(u - u_h, v) = 0$ for all $v \in V_h$ . For $w \in U_h$ this gives $\beta_h \norm{u_h - w} \le \sup_{v} a(u_h - w, v) / \norm{v} = \sup_{v} a(u - w, v) / \norm{v} \le M \norm{u - w}$ . The triangle inequality $\norm{u - u_h} \le \norm{u - w} + \norm{w - u_h}$ then gives the bound.

The case $U = V$ and $U_h = V_h$ is the Galerkin method; there, coercivity with constant $\alpha$ gives $\beta_h \ge \alpha$ . Example: in the streamline upwind Petrov–Galerkin method for $-\varepsilon u'' + u' = f$ , each test function is a trial hat function plus a multiple of its derivative, which adds stabilisation along the flow.

A positive $\beta_h$ is needed: with $\beta_h = 0$ the discrete system is singular, and with $\beta_h \to 0$ along a family of meshes the bound degenerates.

Going deeper

For a student

The theorem is the discrete half of the Babuška theory of problems that are well posed but not coercive. It explains why different test spaces can stabilise convection-dominated problems and why some choices of spaces fail. Its constant $1 + M / \beta_h$ can be improved to $M / \beta_h$ in Hilbert spaces, a result of Xu and Zikatanov. For saddle-point problems, the analogous condition is the Ladyzhenskaya–Babuška–Brezzi condition.

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Tag	`RTBJM23ETS`
URI	`m2://RTBJM23ETS`
Kind	statement, theorem
Writing	W0 stub
MSC	65N30, 65N12
Status	This copy of the site has no build data. Run `m2 status RTBJM23ETS`.
About	Galerkin method
Uses	Hilbert space, Bilinear form, Continuous dual space, Galerkin method