Notation and Conventions

This section records notation used throughout the book.

Typography and indices

Object	Convention	Typical examples
Scalar	italic lowercase or Greek letter	$u$, $T$, $k$, $\rho$
Vector	bold lowercase letter	$\bx$, $\bu$, $\bq$
Second-order tensor	bold uppercase or bold Greek letter	$\bF$, $\bC$, $\bsig$
Fourth-order tensor	blackboard-bold uppercase letter	$\mathbb{C}$
Set or function space	calligraphic or standard space symbol	$\Th$, $\V$, $H^1(\Om)$
Matrix and coefficient vector	bold uppercase letter	$\bK$, $\bU$

When components are useful, Latin indices range over the spatial dimensions and repeated indices are summed unless stated otherwise. Thus

\[ \bm{a}\cdot\bm{b}=a_i b_i, \qquad (\bm{A}\bm{a})_i=A_{ij}a_j, \qquad \bm{A}:\bm{B}=A_{ij}B_{ij}. \]

Geometry and integration

Symbol	Meaning	Symbol	Meaning
$d$	Spatial dimension	$\Om\subset\R^d$	Open physical domain
$\partial\Om$	Boundary of $\Om$	$\Gam_D$, $\Gam_N$	Dirichlet and Neumann portions of the boundary
$\bx$	Point in the spatial or current configuration	$\bX$	Point in a reference configuration
$\bn$	Outward unit normal	$\dx$, $\ds$	Volume and boundary integration measures

Unless a different decomposition is stated,

\[ \partial\Om=\overline{\Gam_D\cup\Gam_N}, \qquad \Gam_D\cap\Gam_N=\varnothing. \]

Differential operators

Symbol	Meaning
$\grad u = \nabla_{\bx} u$	Spatial gradient of a scalar field
$\grad\bu = \nabla_{\bx} \bu$	Spatial gradient of a vector field
$\divg\bq = \nabla_{\bx}\cdot \bq$	Spatial divergence of a vector field
$\divg\bsig = \nabla_{\bx}\cdot\bsig$	Spatial divergence of a second-order tensor field
$\Grad\bu = \nabla_{\bX} \bu$	Gradient with respect to reference coordinates
$\Div\bP = \nabla_{\bX} \cdot \bP$	Divergence with respect to reference coordinates

The gradient of a vector field follows the convention

\[ (\grad\bu)_{ij}=\frac{\partial u_i}{\partial x_j}, \]

and the divergence of a second-order tensor follows

\[ (\divg\bm{A})_i = \frac{\partial A_{ij}}{\partial x_j}. \]

Uppercase operators distinguish differentiation with respect to reference coordinates from differentiation with respect to spatial coordinates.

Function spaces and variational objects

Symbol	Meaning
$L^2(\Om)$	Square-integrable scalar functions on $\Om$
$L^2(\Om;\R^d)$	Square-integrable vector fields on $\Om$
$L^2(\Om;\R^{d\times d})$	Square-integrable second-order tensor fields on $\Om$
$H^m(\Om)$	Functions with square-integrable weak derivatives through order $m$
$H^m(\Om;\R^d)$	Vector fields whose components belong to $H^m(\Om)$
$H_0^1(\Om)$	$H^1$ functions with zero prescribed boundary value
$V$	Generic vector, normed, Banach, or Hilbert space
$\U$	Generic trial space
$\V$	Generic test or variational space
$\Ug$	Trial space satisfying prescribed Dirichlet data $g$
$\Vzero$	Test space satisfying the corresponding homogeneous data
$a(u,v)$	Bilinear form
$a(u;v)$	Semilinear form (nonlinear in $u$ and linear in $v$)
$\ell(v)$	Linear functional
$R(u;v)=\ell(v)-a(u;v)$	Variational residual; use $a(u,v)$ for a bilinear form
$\Pi(u)$	Energy or potential functional
$\delta\Pi(u;v)$	First variation of $\Pi$ at $u$ in direction $v$
$\delta^2\Pi(u;v,w)$	Variation of $\delta\Pi(u;v)$ with respect to $u$ in direction $w$
$\delta_uR(u;v)[w]$	Variation of the residual with respect to $u$, with test function $v$ held fixed
$\norm{v}_V$	Norm of $v$ in $V$
$\seminorm{v}_{H^m(\Om)}$	Sobolev seminorm of order $m$
$\inner{u}{v}_V$	Inner product of $u$ and $v$ in $V$
$\dual{f}{v}$	Duality pairing between a functional $f$ and an argument $v$
$\norm{v}_a=a(v,v)^{1/2}$	Energy norm induced by a symmetric, coercive bilinear form

The subscript identifies the space or inner product associated with the notation. It may be omitted when the choice is clear from the surrounding discussion.

Finite element notation

Symbol	Meaning
$\Th$	Mesh of the domain
$\mathcal K_e\in\Th$	Element $e$ in the mesh
$h_e$	Diameter or characteristic size of element $\mathcal K_e$
$h=\max_{\mathcal K_e\in\Th}h_e$	Global mesh-size parameter
$\rho_e$	Inradius of element $\mathcal K_e$ in shape-regularity estimates
$p$	Polynomial degree of the finite element approximation
$(K,\mathcal P,\mathcal N)$	Element domain, local function space, and degrees of freedom defining a finite element
$\mathbb P_p$, $\mathbb Q_p$	Polynomial spaces of degree $p$ on simplex and tensor-product elements, respectively
$N_e$, $N_v$, $N_n$	Numbers of elements, mesh vertices, and finite element nodes, respectively
$\Uh$, $\Vh$	Discrete trial and test spaces
$\Vhzero$	Discrete test space satisfying homogeneous Dirichlet data
$\uh$, $\vh$	Discrete trial and test functions
$\phi_i$, $\phi_a^e$, $\widehat\phi_a$	Global, element-local, and reference-element basis functions
$\bN^e$	Element shape-function matrix used to interpolate vector-valued fields
$U_i$, $U_a^e$	Global and element-local coefficients
$I^e(a)$	Global number of local degree of freedom $a$ on element $e$
$(I^e)^{-1}(i)$	Local number of global degree of freedom $i$ on element $e$, when $i$ belongs to that element
$x^e$, $\bx^e$	Coordinate map from the reference element $\widehat{\mathcal K}$ to physical element $\mathcal K_e$ in one and several dimensions, respectively
$J_e$, $\bm J_e$	Jacobian of $x^e$ or $\bx^e$ in one and several dimensions, respectively
$\Ie$	Interpolation operator on element $\mathcal K_e$
$\Iref$	Interpolation operator on the reference element
$\Ih$	Global finite element interpolation operator
$C_I$	Mesh-independent constant in an interpolation estimate
$C_{\mathrm{reg}}$	Regularity constant for a dual problem
$\mathcal E(i)$	Set of elements incident on global node or degree of freedom $i$
$\mathcal F_h$	Set of mesh facets
$h_{\mathcal F}$	Diameter or characteristic size of facet $\mathcal F$
$\mathcal Q_h$	Local averaging or quasi-interpolation operator used in residual estimates
$r_e$, $j_{\mathcal F}$	Element and facet residuals
$\eta_e$	Local a posteriori error indicator
$\eta=(\sum_e\eta_e^2)^{1/2}$	Global residual error estimator
$R_i(\bU)=R(u_h;\phi_i)$	Assembled residual entry for a scalar finite element field
$\omega_i$, $\omega_e$, $\omega_{\mathcal F}$	Vertex, element, and facet patches in residual estimates
$C_Q$, $N_{\mathrm{ov}}$	Quasi-interpolation constant and bound on patch overlap
$C_{\mathrm{rel}}$, $C_{\mathrm{eff}}$	Reliability and local-efficiency constants
$\mathrm{osc}_e$	Local oscillation term; its precise definition depends on the stated estimator result
$I_{\mathrm{eff}}=\eta/\norm{u-u_h}_a$	Effectivity index when the exact energy error is known and nonzero
$\mathcal M$, $\theta$	Marked element set and selected fraction of the squared estimator in bulk marking
$\bK^e$, $\bF^e$, $\bU^e$	Element stiffness matrix, load vector, and coefficient vector
$\bK\bU=\bF$	Representative assembled algebraic system
$\bR=\bF-\bK\bU$	Algebraic residual vector for a linear problem, using external minus internal force
$\bm r_p=-\bR_p$	Support reactions at prescribed degrees of freedom
$\bK_T=\partial\bR/\partial\bU$	Jacobian of the nonlinear residual; negative of the internal-force tangent for displacement-independent loads
$\Delta\bU$, $\Delta\bu$	Newton correction to the coefficient vector and displacement field
$\mathbb A_{\alpha I\beta J}=\partial P_{\alpha I}/\partial F_{\beta J}$	Material tangent for the first Piola–Kirchhoff stress
$\zeta$, $\omega$	Load factor and Newton step factor in the nonlinear solution procedure

The book uses element for a geometric member of a mesh. The calligraphic symbols $\Iref$, $\Ie$, and $\Ih$ denote interpolation operators. They are distinct from the connectivity map $I^e$ and the geometric Jacobians $J_e$ and $\bm J_e$.

The symbols $\mathbb P_p$ and $\mathbb Q_p$ denote polynomial spaces. Phrases such as “$P_1$ triangle,” “$P_2$ triangle,” and “$Q_1$ quadrilateral” are the customary names of the corresponding element families.

In finite element formulas, $e$ indexes elements, $a,b$ index local degrees of freedom, and $i,j$ index global degrees of freedom. In continuum component notation, lowercase indices $i,j,k$ denote spatial components, while uppercase indices $I,J,K$ denote material components when reference and current configurations must be distinguished.

The unadorned symbol $h$ is reserved for the mesh-size parameter. The scalar $q=\bq\cdot\bn$ is the outward normal component of $\bq=k\grad u$. In heat conduction, the physical heat-flow vector is $-\bq$, so $q>0$ means that heat enters the body.

Recurring physical fields

Symbol	Meaning	Symbol	Meaning
$T$	Temperature	$\bq=k\grad u$	Flux vector used in the PDE; physical heat flow is $-\bq$
$q=\bq\cdot\bn$	Prescribed normal component; enters $\ell$ with a plus sign	$\bm K$	Conductivity tensor
$\bb$	Body force per unit mass	$\rho\bb$	Body force per unit current volume
$\bb_0$	Body force per unit reference volume	$\rho$	Mass density
$\bu$	Displacement vector	$\beps$	Infinitesimal strain tensor
$\bsig$	Cauchy stress tensor	$\bF$	Deformation gradient
$J = \mathrm{det}(\bF)$	Jacobian (determinant of $\bF$)	$\bC=\bF^T\bF$	Right Cauchy–Green tensor
$I_1 = \mathrm{tr}(\bC)$	First invariant of $\bC$ (trace)	$\bP$	First Piola–Kirchhoff stress tensor
$\bm N$	Outward unit normal in the reference configuration	$\bt$	Prescribed traction vector

Some standard symbols recur in different settings. In finite-deformation mechanics, $\bF$ is the deformation gradient and $J=\det\bF$. In the assembled finite element system, $\bF$ is the load vector, whose entries have nodal and component indices. The surrounding definition and indices distinguish it from the deformation gradient. Similarly, $\bm K$ is a conductivity tensor in the heat equation and $\bK$ is a stiffness matrix in the algebraic system. Their meaning is stated when each setting is introduced.

# Notation and Conventions {#sec-notation .unnumbered} This section records notation used throughout the book. ## Typography and indices | Object | Convention | Typical examples | |---|---|---| | Scalar | italic lowercase or Greek letter | $u$, $T$, $k$, $\rho$ | | Vector | bold lowercase letter | $\bx$, $\bu$, $\bq$ | | Second-order tensor | bold uppercase or bold Greek letter | $\bF$, $\bC$, $\bsig$ | | Fourth-order tensor | blackboard-bold uppercase letter | $\mathbb{C}$ | | Set or function space | calligraphic or standard space symbol | $\Th$, $\V$, $H^1(\Om)$ | | Matrix and coefficient vector | bold uppercase letter | $\bK$, $\bU$ | When components are useful, Latin indices range over the spatial dimensions and repeated indices are summed unless stated otherwise. Thus $$ \bm{a}\cdot\bm{b}=a_i b_i, \qquad (\bm{A}\bm{a})_i=A_{ij}a_j, \qquad \bm{A}:\bm{B}=A_{ij}B_{ij}. $$ ## Geometry and integration | Symbol | Meaning | Symbol | Meaning | |---|---|---|---| | $d$ | Spatial dimension | $\Om\subset\R^d$ | Open physical domain | | $\partial\Om$ | Boundary of $\Om$ | $\Gam_D$, $\Gam_N$ | Dirichlet and Neumann portions of the boundary | | $\bx$ | Point in the spatial or current configuration | $\bX$ | Point in a reference configuration | | $\bn$ | Outward unit normal | $\dx$, $\ds$ | Volume and boundary integration measures | Unless a different decomposition is stated, $$ \partial\Om=\overline{\Gam_D\cup\Gam_N}, \qquad \Gam_D\cap\Gam_N=\varnothing. $$ ## Differential operators | Symbol | Meaning | |---|---| | $\grad u = \nabla_{\bx} u$ | Spatial gradient of a scalar field | | $\grad\bu = \nabla_{\bx} \bu$ | Spatial gradient of a vector field | | $\divg\bq = \nabla_{\bx}\cdot \bq$ | Spatial divergence of a vector field | | $\divg\bsig = \nabla_{\bx}\cdot\bsig$ | Spatial divergence of a second-order tensor field | | $\Grad\bu = \nabla_{\bX} \bu$ | Gradient with respect to reference coordinates | | $\Div\bP = \nabla_{\bX} \cdot \bP$ | Divergence with respect to reference coordinates | The gradient of a vector field follows the convention $$ (\grad\bu)_{ij}=\frac{\partial u_i}{\partial x_j}, $$ and the divergence of a second-order tensor follows $$ (\divg\bm{A})_i = \frac{\partial A_{ij}}{\partial x_j}. $$ Uppercase operators distinguish differentiation with respect to reference coordinates from differentiation with respect to spatial coordinates. ## Function spaces and variational objects | Symbol | Meaning | |---|---| | $L^2(\Om)$ | Square-integrable scalar functions on $\Om$ | | $L^2(\Om;\R^d)$ | Square-integrable vector fields on $\Om$ | | $L^2(\Om;\R^{d\times d})$ | Square-integrable second-order tensor fields on $\Om$ | | $H^m(\Om)$ | Functions with square-integrable weak derivatives through order $m$ | | $H^m(\Om;\R^d)$ | Vector fields whose components belong to $H^m(\Om)$ | | $H_0^1(\Om)$ | $H^1$ functions with zero prescribed boundary value | | $V$ | Generic vector, normed, Banach, or Hilbert space | | $\U$ | Generic trial space | | $\V$ | Generic test or variational space | | $\Ug$ | Trial space satisfying prescribed Dirichlet data $g$ | | $\Vzero$ | Test space satisfying the corresponding homogeneous data | | $a(u,v)$ | Bilinear form | | $a(u;v)$ | Semilinear form (nonlinear in $u$ and linear in $v$) | | $\ell(v)$ | Linear functional | | $R(u;v)=\ell(v)-a(u;v)$ | Variational residual; use $a(u,v)$ for a bilinear form | | $\Pi(u)$ | Energy or potential functional | | $\delta\Pi(u;v)$ | First variation of $\Pi$ at $u$ in direction $v$ | | $\delta^2\Pi(u;v,w)$ | Variation of $\delta\Pi(u;v)$ with respect to $u$ in direction $w$ | | $\delta_uR(u;v)[w]$ | Variation of the residual with respect to $u$, with test function $v$ held fixed | | $\norm{v}_V$ | Norm of $v$ in $V$ | | $\seminorm{v}_{H^m(\Om)}$ | Sobolev seminorm of order $m$ | | $\inner{u}{v}_V$ | Inner product of $u$ and $v$ in $V$ | | $\dual{f}{v}$ | Duality pairing between a functional $f$ and an argument $v$ | | $\norm{v}_a=a(v,v)^{1/2}$ | Energy norm induced by a symmetric, coercive bilinear form | The subscript identifies the space or inner product associated with the notation. It may be omitted when the choice is clear from the surrounding discussion. ## Finite element notation | Symbol | Meaning | |---|---| | $\Th$ | Mesh of the domain | | $\mathcal K_e\in\Th$ | Element $e$ in the mesh | | $h_e$ | Diameter or characteristic size of element $\mathcal K_e$ | | $h=\max_{\mathcal K_e\in\Th}h_e$ | Global mesh-size parameter | | $\rho_e$ | Inradius of element $\mathcal K_e$ in shape-regularity estimates | | $p$ | Polynomial degree of the finite element approximation | | $(K,\mathcal P,\mathcal N)$ | Element domain, local function space, and degrees of freedom defining a finite element | | $\mathbb P_p$, $\mathbb Q_p$ | Polynomial spaces of degree $p$ on simplex and tensor-product elements, respectively | | $N_e$, $N_v$, $N_n$ | Numbers of elements, mesh vertices, and finite element nodes, respectively | | $\Uh$, $\Vh$ | Discrete trial and test spaces | | $\Vhzero$ | Discrete test space satisfying homogeneous Dirichlet data | | $\uh$, $\vh$ | Discrete trial and test functions | | $\phi_i$, $\phi_a^e$, $\widehat\phi_a$ | Global, element-local, and reference-element basis functions | | $\bN^e$ | Element shape-function matrix used to interpolate vector-valued fields | | $U_i$, $U_a^e$ | Global and element-local coefficients | | $I^e(a)$ | Global number of local degree of freedom $a$ on element $e$ | | $(I^e)^{-1}(i)$ | Local number of global degree of freedom $i$ on element $e$, when $i$ belongs to that element | | $x^e$, $\bx^e$ | Coordinate map from the reference element $\widehat{\mathcal K}$ to physical element $\mathcal K_e$ in one and several dimensions, respectively | | $J_e$, $\bm J_e$ | Jacobian of $x^e$ or $\bx^e$ in one and several dimensions, respectively | | $\Ie$ | Interpolation operator on element $\mathcal K_e$ | | $\Iref$ | Interpolation operator on the reference element | | $\Ih$ | Global finite element interpolation operator | | $C_I$ | Mesh-independent constant in an interpolation estimate | | $C_{\mathrm{reg}}$ | Regularity constant for a dual problem | | $\mathcal E(i)$ | Set of elements incident on global node or degree of freedom $i$ | | $\mathcal F_h$ | Set of mesh facets | | $h_{\mathcal F}$ | Diameter or characteristic size of facet $\mathcal F$ | | $\mathcal Q_h$ | Local averaging or quasi-interpolation operator used in residual estimates | | $r_e$, $j_{\mathcal F}$ | Element and facet residuals | | $\eta_e$ | Local a posteriori error indicator | | $\eta=(\sum_e\eta_e^2)^{1/2}$ | Global residual error estimator | | $R_i(\bU)=R(u_h;\phi_i)$ | Assembled residual entry for a scalar finite element field | | $\omega_i$, $\omega_e$, $\omega_{\mathcal F}$ | Vertex, element, and facet patches in residual estimates | | $C_Q$, $N_{\mathrm{ov}}$ | Quasi-interpolation constant and bound on patch overlap | | $C_{\mathrm{rel}}$, $C_{\mathrm{eff}}$ | Reliability and local-efficiency constants | | $\mathrm{osc}_e$ | Local oscillation term; its precise definition depends on the stated estimator result | | $I_{\mathrm{eff}}=\eta/\norm{u-u_h}_a$ | Effectivity index when the exact energy error is known and nonzero | | $\mathcal M$, $\theta$ | Marked element set and selected fraction of the squared estimator in bulk marking | | $\bK^e$, $\bF^e$, $\bU^e$ | Element stiffness matrix, load vector, and coefficient vector | | $\bK\bU=\bF$ | Representative assembled algebraic system | | $\bR=\bF-\bK\bU$ | Algebraic residual vector for a linear problem, using external minus internal force | | $\bm r_p=-\bR_p$ | Support reactions at prescribed degrees of freedom | | $\bK_T=\partial\bR/\partial\bU$ | Jacobian of the nonlinear residual; negative of the internal-force tangent for displacement-independent loads | | $\Delta\bU$, $\Delta\bu$ | Newton correction to the coefficient vector and displacement field | | $\mathbb A_{\alpha I\beta J}=\partial P_{\alpha I}/\partial F_{\beta J}$ | Material tangent for the first Piola--Kirchhoff stress | | $\zeta$, $\omega$ | Load factor and Newton step factor in the nonlinear solution procedure | The book uses *element* for a geometric member of a mesh. The calligraphic symbols $\Iref$, $\Ie$, and $\Ih$ denote interpolation operators. They are distinct from the connectivity map $I^e$ and the geometric Jacobians $J_e$ and $\bm J_e$. The symbols $\mathbb P_p$ and $\mathbb Q_p$ denote polynomial spaces. Phrases such as “$P_1$ triangle,” “$P_2$ triangle,” and “$Q_1$ quadrilateral” are the customary names of the corresponding element families. In finite element formulas, $e$ indexes elements, $a,b$ index local degrees of freedom, and $i,j$ index global degrees of freedom. In continuum component notation, lowercase indices $i,j,k$ denote spatial components, while uppercase indices $I,J,K$ denote material components when reference and current configurations must be distinguished. The unadorned symbol $h$ is reserved for the mesh-size parameter. The scalar $q=\bq\cdot\bn$ is the outward normal component of $\bq=k\grad u$. In heat conduction, the physical heat-flow vector is $-\bq$, so $q>0$ means that heat enters the body. ## Recurring physical fields | Symbol | Meaning | Symbol | Meaning | |---|---|---|---| | $T$ | Temperature | $\bq=k\grad u$ | Flux vector used in the PDE; physical heat flow is $-\bq$ | | $q=\bq\cdot\bn$ | Prescribed normal component; enters $\ell$ with a plus sign | $\bm K$ | Conductivity tensor | | $\bb$ | Body force per unit mass | $\rho\bb$ | Body force per unit current volume | | $\bb_0$ | Body force per unit reference volume | $\rho$ | Mass density | | $\bu$ | Displacement vector | $\beps$ | Infinitesimal strain tensor | | $\bsig$ | Cauchy stress tensor | $\bF$ | Deformation gradient | | $J = \mathrm{det}(\bF)$ | Jacobian (determinant of $\bF$) | $\bC=\bF^T\bF$ | Right Cauchy--Green tensor | | $I_1 = \mathrm{tr}(\bC)$ | First invariant of $\bC$ (trace) | $\bP$ | First Piola--Kirchhoff stress tensor | | $\bm N$ | Outward unit normal in the reference configuration | $\bt$ | Prescribed traction vector | Some standard symbols recur in different settings. In finite-deformation mechanics, $\bF$ is the deformation gradient and $J=\det\bF$. In the assembled finite element system, $\bF$ is the load vector, whose entries have nodal and component indices. The surrounding definition and indices distinguish it from the deformation gradient. Similarly, $\bm K$ is a conductivity tensor in the heat equation and $\bK$ is a stiffness matrix in the algebraic system. Their meaning is stated when each setting is introduced.

Symbol	Meaning
\(L^2(\Om)\)	Square-integrable scalar functions on \(\Om\)
\(L^2(\Om;\R^d)\)	Square-integrable vector fields on \(\Om\)
\(L^2(\Om;\R^{d\times d})\)	Square-integrable second-order tensor fields on \(\Om\)
\(H^m(\Om)\)	Functions with square-integrable weak derivatives through order \(m\)
\(H^m(\Om;\R^d)\)	Vector fields whose components belong to \(H^m(\Om)\)
\(H_0^1(\Om)\)	\(H^1\) functions with zero prescribed boundary value
\(V\)	Generic vector, normed, Banach, or Hilbert space
\(\U\)	Generic trial space
\(\V\)	Generic test or variational space
\(\Ug\)	Trial space satisfying prescribed Dirichlet data \(g\)
\(\Vzero\)	Test space satisfying the corresponding homogeneous data
\(a(u,v)\)	Bilinear form
\(a(u;v)\)	Semilinear form (nonlinear in \(u\) and linear in \(v\))
\(\ell(v)\)	Linear functional
\(R(u;v)=\ell(v)-a(u;v)\)	Variational residual; use \(a(u,v)\) for a bilinear form
\(\Pi(u)\)	Energy or potential functional
\(\delta\Pi(u;v)\)	First variation of \(\Pi\) at \(u\) in direction \(v\)
\(\delta^2\Pi(u;v,w)\)	Variation of \(\delta\Pi(u;v)\) with respect to \(u\) in direction \(w\)
\(\delta_uR(u;v)[w]\)	Variation of the residual with respect to \(u\), with test function \(v\) held fixed
\(\norm{v}_V\)	Norm of \(v\) in \(V\)
\(\seminorm{v}_{H^m(\Om)}\)	Sobolev seminorm of order \(m\)
\(\inner{u}{v}_V\)	Inner product of \(u\) and \(v\) in \(V\)
\(\dual{f}{v}\)	Duality pairing between a functional \(f\) and an argument \(v\)
\(\norm{v}_a=a(v,v)^{1/2}\)	Energy norm induced by a symmetric, coercive bilinear form

Symbol	Meaning
\(\Th\)	Mesh of the domain
\(\mathcal K_e\in\Th\)	Element \(e\) in the mesh
\(h_e\)	Diameter or characteristic size of element \(\mathcal K_e\)
\(h=\max_{\mathcal K_e\in\Th}h_e\)	Global mesh-size parameter
\(\rho_e\)	Inradius of element \(\mathcal K_e\) in shape-regularity estimates
\(p\)	Polynomial degree of the finite element approximation
\((K,\mathcal P,\mathcal N)\)	Element domain, local function space, and degrees of freedom defining a finite element
\(\mathbb P_p\), \(\mathbb Q_p\)	Polynomial spaces of degree \(p\) on simplex and tensor-product elements, respectively
\(N_e\), \(N_v\), \(N_n\)	Numbers of elements, mesh vertices, and finite element nodes, respectively
\(\Uh\), \(\Vh\)	Discrete trial and test spaces
\(\Vhzero\)	Discrete test space satisfying homogeneous Dirichlet data
\(\uh\), \(\vh\)	Discrete trial and test functions
\(\phi_i\), \(\phi_a^e\), \(\widehat\phi_a\)	Global, element-local, and reference-element basis functions
\(\bN^e\)	Element shape-function matrix used to interpolate vector-valued fields
\(U_i\), \(U_a^e\)	Global and element-local coefficients
\(I^e(a)\)	Global number of local degree of freedom \(a\) on element \(e\)
\((I^e)^{-1}(i)\)	Local number of global degree of freedom \(i\) on element \(e\), when \(i\) belongs to that element
\(x^e\), \(\bx^e\)	Coordinate map from the reference element \(\widehat{\mathcal K}\) to physical element \(\mathcal K_e\) in one and several dimensions, respectively
\(J_e\), \(\bm J_e\)	Jacobian of \(x^e\) or \(\bx^e\) in one and several dimensions, respectively
\(\Ie\)	Interpolation operator on element \(\mathcal K_e\)
\(\Iref\)	Interpolation operator on the reference element
\(\Ih\)	Global finite element interpolation operator
\(C_I\)	Mesh-independent constant in an interpolation estimate
\(C_{\mathrm{reg}}\)	Regularity constant for a dual problem
\(\mathcal E(i)\)	Set of elements incident on global node or degree of freedom \(i\)
\(\mathcal F_h\)	Set of mesh facets
\(h_{\mathcal F}\)	Diameter or characteristic size of facet \(\mathcal F\)
\(\mathcal Q_h\)	Local averaging or quasi-interpolation operator used in residual estimates
\(r_e\), \(j_{\mathcal F}\)	Element and facet residuals
\(\eta_e\)	Local a posteriori error indicator
\(\eta=(\sum_e\eta_e^2)^{1/2}\)	Global residual error estimator
\(R_i(\bU)=R(u_h;\phi_i)\)	Assembled residual entry for a scalar finite element field
\(\omega_i\), \(\omega_e\), \(\omega_{\mathcal F}\)	Vertex, element, and facet patches in residual estimates
\(C_Q\), \(N_{\mathrm{ov}}\)	Quasi-interpolation constant and bound on patch overlap
\(C_{\mathrm{rel}}\), \(C_{\mathrm{eff}}\)	Reliability and local-efficiency constants
\(\mathrm{osc}_e\)	Local oscillation term; its precise definition depends on the stated estimator result
\(I_{\mathrm{eff}}=\eta/\norm{u-u_h}_a\)	Effectivity index when the exact energy error is known and nonzero
\(\mathcal M\), \(\theta\)	Marked element set and selected fraction of the squared estimator in bulk marking
\(\bK^e\), \(\bF^e\), \(\bU^e\)	Element stiffness matrix, load vector, and coefficient vector
\(\bK\bU=\bF\)	Representative assembled algebraic system
\(\bR=\bF-\bK\bU\)	Algebraic residual vector for a linear problem, using external minus internal force
\(\bm r_p=-\bR_p\)	Support reactions at prescribed degrees of freedom
\(\bK_T=\partial\bR/\partial\bU\)	Jacobian of the nonlinear residual; negative of the internal-force tangent for displacement-independent loads
\(\Delta\bU\), \(\Delta\bu\)	Newton correction to the coefficient vector and displacement field
\(\mathbb A_{\alpha I\beta J}=\partial P_{\alpha I}/\partial F_{\beta J}\)	Material tangent for the first Piola–Kirchhoff stress
\(\zeta\), \(\omega\)	Load factor and Newton step factor in the nonlinear solution procedure

Symbol	Meaning	Symbol	Meaning
\(T\)	Temperature	\(\bq=k\grad u\)	Flux vector used in the PDE; physical heat flow is \(-\bq\)
\(q=\bq\cdot\bn\)	Prescribed normal component; enters \(\ell\) with a plus sign	\(\bm K\)	Conductivity tensor
\(\bb\)	Body force per unit mass	\(\rho\bb\)	Body force per unit current volume
\(\bb_0\)	Body force per unit reference volume	\(\rho\)	Mass density
\(\bu\)	Displacement vector	\(\beps\)	Infinitesimal strain tensor
\(\bsig\)	Cauchy stress tensor	\(\bF\)	Deformation gradient
\(J = \mathrm{det}(\bF)\)	Jacobian (determinant of \(\bF\))	\(\bC=\bF^T\bF\)	Right Cauchy–Green tensor
\(I_1 = \mathrm{tr}(\bC)\)	First invariant of \(\bC\) (trace)	\(\bP\)	First Piola–Kirchhoff stress tensor
\(\bm N\)	Outward unit normal in the reference configuration	\(\bt\)	Prescribed traction vector