Variational crimes and Strang lemmas

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.

Let $V$ be a Hilbert space and let $V_h\subseteq V$ be a subspace. Suppose the exact form $a$ is coercive with constant $\alpha >0$ in the norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_V$. Let $a_h$ be a discrete form on $V_h$ and suppose

$\mathrm{\mid }{a}_h(v_h,v_h)-a(v_h,v_h)\mathrm{\mid }\le {\eta }_h\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_V^2(v_h\in V_h)$

for some ${\eta }_h\ge 0$. Then $a_h$ is coercive with constant at least $\alpha -{\eta }_h$ whenever ${\eta }_h<\alpha$. In particular, if ${\eta }_h\le \alpha \mathrm{/}2$, then its coercivity constant is at least $\alpha \mathrm{/}2$.

Assume Countable Choice. Let $V$ be a real or complex Hilbert space, $a:V\times V\to \mathbb{K}$ a bounded coercive form with boundedness constant $M$ and coercivity constant $\alpha >0$, and $F:V\to \mathbb{K}$ a bounded functional linear in the real case and conjugate-linear in the complex test argument. Let $V_h\subseteq V$ be finite-dimensional, and let $a_h:V_h\times V_h\to \mathbb{K}$ and $F_h:V_h\to \mathbb{K}$ be bounded, with $F_h$ having the same test-linearity as $F$, and $a_h$ coercive with constant ${\alpha }_h>0$ in the inherited norm. Both $a$ and $a_h$ are sesquilinear (bilinear over $\mathbb{R}$), linear in their first argument and conjugate-linear in their second over $\mathbb{C}$. There are unique exact and discrete solutions $u$ and $u_h$, and

$\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_V\le {\inf }_{v_h\in V_h}((1+M\mathrm{/}{\alpha }_h)\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_V+\frac{{\delta }_{a,h}(v_h)+{\delta }_{F,h}}{{\alpha }_h}),$

where the defects are those of Variational-crime settings and consistency defects. For a family of meshes this is a uniform estimate whenever ${\inf }_h{\alpha }_h>0$ and the displayed approximation and defect terms are uniformly controlled. The term ${\delta }_{F,h}$ retains source approximation and data oscillation.

Assume Countable Choice. Let $V$ be a real or complex Hilbert space, $a:V\times V\to \mathbb{K}$ a bounded coercive sesquilinear form (bilinear over $\mathbb{R}$), and $F$ a bounded functional with the test-linearity required by $a$. Let $V_h$ be a finite-dimensional Hilbert space with norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_h$, not necessarily a subspace of $V$. Suppose the algebraic sum $X_h=V+V_h$ is equipped with a norm extending these norms, and ${\tilde{a}}_h:X_h\times V_h\to \mathbb{K}$ is a bounded sesquilinear extension of a discrete form $a_h$ with bound $M_h$. Over $\mathbb{C}$, both forms are linear in the first argument and conjugate-linear in the second; over $\mathbb{R}$, they are bilinear. Assume $a_h$ is coercive on $V_h$ with constant ${\alpha }_h>0$, and let $F_h$ be bounded on $V_h$ with the matching test-linearity. The exact and discrete equations have unique solutions $u$ and $u_h$, and

$\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_h\le (1+M_h\mathrm{/}{\alpha }_h){\inf }_{v_h\in V_h}\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_h+{\delta }_h^{\mathrm{n}\mathrm{c}}(u)\mathrm{/}{\alpha }_h,$

where ${\delta }_h^{\mathrm{n}\mathrm{c}}(u)={\sup }_{w_h\in V_h\setminus \{0\}}\mathrm{\mid }{F}_h(w_h)-{\tilde{a}}_h(u,w_h)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_h$ when $V_h\mathrm{\ne }\{0\}$, and ${\delta }_h^{\mathrm{n}\mathrm{c}}(u):=0$ when $V_h=\{0\}$. The estimate requires the bounded extension and does not assert that every nonconforming, penalty, or discontinuous Galerkin form has one.

For a family of these problems, suppose $M_h\le \bar{M}<\mathrm{\infty }$ and ${\alpha }_h\ge \underset{‾}{\alpha }>0$ uniformly. Writing $E_h={\inf }_{v_h\in V_h}\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_h$, the uniform bound is $\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_h\le (1+\bar{M}\mathrm{/}\underset{‾}{\alpha })E_h+{\delta }_h^{\mathrm{n}\mathrm{c}}(u)\mathrm{/}\underset{‾}{\alpha }\mathrm{.}$ Thus uniform control of $E_h$ and ${\delta }_h^{\mathrm{n}\mathrm{c}}(u)$ gives a uniform error budget; if both tend to zero, the errors tend to zero, and if both are $O(h^p)$ for a fixed $p>0$, the error is $O(h^p)$. No rate is inferred without those approximation, residual, and stability bounds.

Assume Countable Choice. Let ${\mathrm{\Omega }}_h,\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be open, and let ${\mathrm{\Phi }}_h:{\mathrm{\Omega }}_h\to \mathrm{\Omega }$ be a $C^1$ diffeomorphism. Write $J_h(y):=\mathrm{\mid }\det D{\mathrm{\Phi }}_h(y)\mathrm{\mid }>0$. Let $A:\mathrm{\Omega }\to {\mathbb{R}}^{d\times d}$ be Lebesgue measurable, real symmetric almost everywhere, and satisfy $\lambda I⪯A(x)⪯\mathrm{\Lambda }I$ almost everywhere, for constants $0<\lambda \le \mathrm{\Lambda }<\mathrm{\infty }$. The physical exact form on real-valued functions with square-integrable gradients is $a(U,W):={\int }_{\mathrm{\Omega }}\mathrm{\nabla }U(x{)}^{T}A(x)\mathrm{\nabla }W(x)dx$. The pulled-back exact tensor and form are

$B_h(y):=J_h(y)D{\mathrm{\Phi }}_h(y{)}^{-1}A({\mathrm{\Phi }}_h(y))D{\mathrm{\Phi }}_h(y{)}^{-T},a_h^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}(v,w):={\int }_{{\mathrm{\Omega }}_h}\mathrm{\nabla }v(y{)}^T{B}_h(y)\mathrm{\nabla }w(y)dy\mathrm{.}$

The lifted form is defined for real-valued $v,w$ with measurable gradients and finite pulled-back energies ${\int }_{{\mathrm{\Omega }}_h}\mathrm{\nabla }{v}^T{B}_h\mathrm{\nabla }v<\mathrm{\infty }$ and ${\int }_{{\mathrm{\Omega }}_h}\mathrm{\nabla }{w}^T{B}_h\mathrm{\nabla }w<\mathrm{\infty }$. The pointwise Cauchy–Schwarz inequality for the positive tensor $B_h$ bounds the absolute cross density by the product of the square roots of these energy densities, which is integrable by Holder's inequality for integrals, including the endpoint cases. Square-integrability of $\mathrm{\nabla }v,\mathrm{\nabla }w$ alone does not guarantee these energy conditions for an arbitrary diffeomorphism.

The geometry tensor discrepancy of an implemented tensor ${\tilde{B}}_h$ on a cell $K\subset {\mathrm{\Omega }}_h$ is ${\eta }_{G,K}:=\mathrm{\parallel }{B}_h-{\tilde{B}}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K;\mathrm{o}\mathrm{p})}$, where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ is the Euclidean operator norm. The geometry-lifted form defect is the difference between $a_h^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}$ and the form using ${\tilde{B}}_h$, on pairs for which both displayed cross densities are absolutely integrable. No meshwise approximation rate is part of this definition.

Assume the geometry setting of Geometry lift and transformed elliptic tensor, and suppose ${\mathrm{\Omega }}_h$ has a nonempty finite measurable partition into bounded cells $\{K\}$, with $h_K:=\mathrm{diam}(K)$ and $h:={\max }_K{h}_K$. For real-valued $U,W\in C^1(\mathrm{\Omega })$ with $\mathrm{\nabla }U,\mathrm{\nabla }W\in L^2(\mathrm{\Omega })$, put $v:=U\circ {\mathrm{\Phi }}_h$ and $w:=W\circ {\mathrm{\Phi }}_h$, and assume their gradients are square-integrable on each cell. Then the physical exact form $a(U,W)$ pulled back by ${\mathrm{\Phi }}_h$ is $a_h^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}(v,w)$. Suppose the implemented tensors ${\tilde{B}}_h$ are real symmetric and measurable, with ${\eta }_{G,K}<\mathrm{\infty }$. The form ${\tilde{a}}_h(v,w):={\sum }_K{\int }_K\mathrm{\nabla }{v}^T{\tilde{B}}_h\mathrm{\nabla }w$ satisfies

$\mathrm{\mid }{a}_h^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}(v,w)-{\tilde{a}}_h(v,w)\mathrm{\mid }\le {\sum }_K{\eta }_{G,K}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(K)}\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(K)}\le {\eta }_G\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_h)}\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_h)},$

where ${\eta }_G:={\max }_K{\eta }_{G,K}$. If ${\eta }_{G,K}\le C_G{h}_K^s$ for $s>0$, the form defect is at most $C_G{h}^s\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2$. Moreover, if $J_h\ge j_{-}>0$, $0<L$, and $\mathrm{\parallel }D{\mathrm{\Phi }}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }},\mathrm{o}\mathrm{p}}\le L$, then $B_h⪰\beta I$ with $\beta =j_{-}\lambda \mathrm{/}{L}^2$. For the stability conclusion, let $Z_h$ be a real Hilbert space of pullback functions with inner product ${\int }_{{\mathrm{\Omega }}_h}\mathrm{\nabla }v\cdot \mathrm{\nabla }w$, so its norm is $\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_h)}$. Restrict the exact and implemented forms to $Z_h$. If ${\eta }_G\le \beta \mathrm{/}2$, the implemented form is coercive on $Z_h$ with constant at least $\beta \mathrm{/}2$. The rate assumption is separate from the lift; neither it nor completeness of the gradient-norm space follows from a lift alone.

Assume Countable Choice and the exact conforming Galerkin hypotheses of Hermitian Galerkin solutions are best in the energy norm, with Hermitian form $a$ bounded by $M$ in $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_V$. Let $u_h^G$ be the exact Galerkin solution. Let $a_h$ be a bounded discrete form on the same $V_h$, coercive in $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_V$ with constant ${\alpha }_h>0$, and let $F_h$ be a bounded discrete load with the matching test-linearity. Its unique perturbed solution ${\tilde{u}}_h$ satisfies

$\mathrm{\parallel }u-{\tilde{u}}_h{\mathrm{\parallel }}_a\le {\inf }_{v_h\in V_h}\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_a+\frac{\sqrt{M}}{{\alpha }_h}{\rho }_h,$

where ${\rho }_h:={\sup }_{w_h\mathrm{\ne }0}\mathrm{\mid }{F}_h(w_h)-a_h(u_h^G,w_h)\mathrm{\mid }\mathrm{/}\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_V$, defined as zero if $V_h=\{0\}$. This uses the exact Galerkin solution as a baseline; ${\rho }_h$ must be bounded from the actual form and load defects.

Assume Countable Choice. Let $0=x_0<x_1<\cdots <x_N=1$ be a finite interval partition, $h:={\max }_j(x_{j+1}-x_j)$, and $u\in C^2([0,1])$. Let $I_{h}u$ be the continuous piecewise-linear function with nodal values $I_{h}u(x_j)=u(x_j)$. Then

$\mathrm{\parallel }(u-I_{h}u{)}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(0,1)}\le h\mathrm{\parallel }{u}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(0,1)}\mathrm{.}$

If $u(0)=u(1)=0$, then $I_{h}u$ also vanishes at both endpoints.

Assume Countable Choice. For every real $v\in H_0^1(0,1)$,

$\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(0,1)}\le \mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(0,1)}\mathrm{.}$

Consequently $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1(0,1)}\le \sqrt{2}\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(0,1)}$.