Variational-crime settings and consistency defects

Let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. In the exact variational problem, $V$ is a Hilbert space, $a:V\times V\to \mathbb{K}$ is a bounded coercive sesquilinear form (bilinear when $\mathbb{K}=\mathbb{R}$), linear in its first argument and conjugate-linear in its second when $\mathbb{K}=\mathbb{C}$, and $F:V\to \mathbb{K}$ is bounded and has the same linearity in its test argument as the second argument of $a$. Exact solutions satisfy $a(u,v)=F(v)$ for every $v\in V$. Boundedness and coercivity have the meanings in Bounded, coercive and symmetric sesquilinear forms.

For a conforming perturbed Galerkin problem, $V_h\subseteq V$ is a finite-dimensional subspace, $a_h:V_h\times V_h\to \mathbb{K}$ and $F_h:V_h\to \mathbb{K}$ are the discrete form and load, and $u_h\in V_h$ solves $a_h(u_h,w_h)=F_h(w_h)$. Its discrete coercivity constant is any ${\alpha }_h>0$ such that $\mathrm{Re}{a}_h(w_h,w_h)\ge {\alpha }_h\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_V^2$ for every $w_h\in V_h$. For $v_h\in V_h$, define

${\delta }_{a,h}(v_h):={\sup }_{w_h\in V_h\setminus \{0\}}\frac{\mathrm{\mid }a(v_h,w_h)-a_h(v_h,w_h)\mathrm{\mid }}{\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_V},{\delta }_{F,h}:={\sup }_{w_h\in V_h\setminus \{0\}}\frac{\mathrm{\mid }F(w_h)-F_h(w_h)\mathrm{\mid }}{\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_V}\mathrm{.}$

If $V_h=\{0\}$, both defects are defined to be zero. The load defect ${\delta }_{F,h}$ includes any source approximation or data oscillation; it is not assumed small or zero without a separate bound.

For a nonconforming problem, the exact space $V$ and finite-dimensional trial space $V_h$ need not satisfy $V_h\subseteq V$. Put $X_h:=V+V_h$ and equip it with a norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_h$ whose restrictions to $V$ and $V_h$ are norms. The discrete form $a_h$ is defined on $V_h\times V_h$, and a bounded extension ${\tilde{a}}_h:X_h\times V_h\to \mathbb{K}$ means ${\tilde{a}}_h=a_h$ on $V_h\times V_h$ and $\mathrm{\mid }{\tilde{a}}_h(z,w_h)\mathrm{\mid }\le M_h\mathrm{\parallel }z{\mathrm{\parallel }}_h\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_h$. For an exact solution $u\in V$, define the nonconforming consistency defect

${\delta }_h^{\mathrm{n}\mathrm{c}}(u):={\sup }_{w_h\in V_h\setminus \{0\}}\frac{\mathrm{\mid }{F}_h(w_h)-{\tilde{a}}_h(u,w_h)\mathrm{\mid }}{\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_h},$

with value zero if $V_h=\{0\}$. This residual need not vanish: the exact weak equation is only asserted for tests in $V$.

Geometry and quadrature defects are components of the appropriate form defect when the implemented form is compared with the pulled-back exact form. No exact Galerkin orthogonality is asserted after replacing the form or test space.