= Petrov-Galerkin method
{c}
{title2=$a(u_h,v_h)=\ell(v_h)\quad(v_h\in W_h)$}
{wiki=Petrov–Galerkin_method}
The <Petrov-Galerkin method> chooses an approximate solution $u_h$ in a finite-dimensional trial <vector space> $V_h$ and tests a <weak formulation> against a possibly different test space $W_h$. For a <bilinear form> $a$ and a <linear functional> $\ell$, it imposes $a(u_h,v_h)=\ell(v_h)$ for every $v_h\in W_h$. Taking $W_h=V_h$ 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>.
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