The Petrov–Galerkin method is a numerical technique used to solve partial differential equations (PDEs), primarily in the context of finite element analysis. It is a variant of the Galerkin method, which is widely used for approximating solutions to boundary value problems.
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The Petrov-Galerkin method chooses an approximate solution in a finite-dimensional trial vector space and tests a weak formulation against a possibly different test space . For a bilinear form and a linear functional , it imposes for every . Taking recovers the usual Galerkin method. Different test functions can alter stability and accuracy; the Galerkin orthogonality is then against the test space and need not be an orthogonal projection in an energy norm.