PetrovGalerkin

Petrov-Galerkin is a family of numerical methods used to discretize partial differential equations by generalizing the Galerkin approach. Unlike the standard Galerkin method, which uses the same function space for trial solutions and test functions, the Petrov-Galerkin approach employs distinct spaces for the trial (solution) space and the test (weighting) space. This asymmetry can improve stability and accuracy for certain problem classes.

In abstract form, let a(u, v) be a bilinear form and f(v) a linear functional on a pair of spaces V and W, with the goal of finding u in V such that a(u, w) = f(w) for all w in W. The choice of V and W is problem dependent; the test space W is often designed to stabilize the numerical scheme or to target particular features of the solution. For specific PDEs, this leads to different stabilization strategies while preserving a well-posed discrete problem.

Variants of the Petrov-Galerkin method include streamline upwind Petrov-Galerkin (SUPG), which introduces upwind-like weighting to stabilize advection-dominated problems; discontinuous Petrov-Galerkin (DPG), which uses broken test spaces and aims for automatically stable formulations with controllable error estimates; and least-squares Petrov-Galerkin (LSPG), used in reduced-order modeling and nonlinear contexts.

Key considerations include ensuring stability through appropriate inf-sup (Ladyzhenskaya–Babuška–Brezzi) conditions for the chosen spaces, and managing the potentially increased cost of assembling and solving the resulting systems. The method finds wide use in fluid dynamics, convection–diffusion problems, electromagnetics, and structural analysis, often within finite element frameworks.