Appendix 1: Calculation of deformation gradient for FCC cell
Here we discuss computational expressions for continuous deformation field and deformation gradient when FCC cubic cell (hereinafter referred to as FCC cell) is used as basic deformation element.
Consider the standard cube \({\mathscr {Q}}=[-1,1]^3\). Corresponding to FCC cell, the 8 vertexes are denoted by \(P_1,\ldots ,P_8\), and the 6 face center points are denoted by \(P_9,\ldots ,P_{14}\) (as shown in Fig. 5).
Denote the coordinate of \(P_i\) by \((p_{i1},p_{i2},p_{i3})\quad (i=1,\ldots ,14)\), then
$$\begin{aligned}&(p_{i1},p_{i2},p_{i3})\nonumber \\&\quad = {\left\{ \begin{array}{ll} (\pm 1,\pm 1,\pm 1) &{} i=1,\ldots ,8 \\ (\pm 1,0,0),(0,\pm 1,0) \text { or } (0,0,\pm 1) &{} p=9,\ldots ,14 \end{array}\right. }\nonumber \\ \end{aligned}$$
(54)
For \({\xi }=(\xi _1,\xi _2,\xi _3)\in {\mathscr {Q}}\), let
$$\begin{aligned}&\phi _i(\xi _1,\xi _2,\xi _3)= {\left\{ \begin{array}{ll} \frac{1}{8}{\prod }_{j=1}^3(1+p_{ij}\xi _j)- \frac{1}{8}\sum _{k=1}^3{\prod }_{j=1}^3(1-\xi _j^2)/(1-p_{ik}\xi _k) &{} \quad i = 1,\ldots ,8\\ \frac{1}{2}{\prod }_{j=1}^3(1-\xi _j^2)/{\prod }_{j=1}^3(1+p_{ij}\xi _j) &{}\quad i = 9,\ldots ,14 \end{array}\right. } \end{aligned}$$
(55)
then
$$\begin{aligned} \phi _i(P_j)=\delta _{ij}= {\left\{ \begin{array}{ll} 1 &{} i=j\\ 0 &{} i\ne j \end{array}\right. } \quad (i,j=1,\ldots ,14) \end{aligned}$$
(56)
Thus, \({\phi _1,\ldots ,\phi _{14}}\) are linearly independent, and they constitute a set of interpolation basis functions on \({\mathscr {Q}}\).
For any function \(u(\xi _1,\xi _2,\xi _3)\) on \({\mathscr {Q}}\), define the interpolation operator \({\mathscr {I}}\) as
$$\begin{aligned} {\mathscr {I}}u(\xi )=\sum _{i=1}^{14} u(P_i)\phi _i(\xi ). \end{aligned}$$
(57)
Just like in the BCC case, \({\mathscr {I}}\) keeps first-degree polynomials unchanged.
Now suppose that \(\varOmega _r\) is a FCC cell with edge length 2h, and the volume center of the cell is at \(X^C\). Then we have an affine transformation T that maps \(\varOmega _r\) to the standard cube \({\mathscr {Q}}\) (as shown in Fig. 6):
$$\begin{aligned} \xi =T(X)=\frac{X-X^C}{h}. \end{aligned}$$
(58)
For any function f(X) on \(\varOmega _r\), define the interpolation operator \({\mathscr {I}}_h\) as
$$\begin{aligned} {\mathscr {I}}_h f(X)= & {} \sum _{i=1}^{14} f(T^{-1}(P_i))\phi _i(T(X))\nonumber \\= & {} \sum _{i=1}^{14} f(X_i)\phi _i\left( \frac{X-X^C}{h}\right) , \end{aligned}$$
(59)
where \(X_i (i=1,\ldots ,14)\) are the undeformed positions of the 14 atoms in the cell.
Using \({\mathscr {I}}_h\), the deformed position \(x=R(X)\) of any \(X\in \varOmega _r\) can be expressed by the deformed positions of the lattice points \(x_i (i=1,\ldots ,14)\):
$$\begin{aligned} {\mathscr {I}}_hR(X)= & {} \sum _{i=1}^{14} R(X_i)\phi _i\left( \frac{X-X^C}{h}\right) \nonumber \\= & {} \sum _{i=1}^{14} x_i\phi _i\left( \frac{X-X^C}{h}\right) . \end{aligned}$$
(60)
Furthermore, the deformation gradient at X can be approximately expressed as
$$\begin{aligned} \tilde{F}(X) =\frac{\partial {\mathscr {I}}_hR(X)}{\partial X} =\sum _{i=1}^{14} \frac{x_i}{h}\otimes \nabla \phi _i\left( \frac{X-X^C}{h}\right) . \end{aligned}$$
(61)
Appendix 2: Detailed derivation of error estimate
Using Eq. (38) we have
$$\begin{aligned}&\left| \int _{\varOmega _r}\left( \nabla \cdot {P}(X)- \nabla \cdot \tilde{P}(X)\right) {{\mathrm{d}}}X \right| \nonumber \\&\quad \le \frac{1}{V_r}\sum _{\alpha }\sum _{\beta \not =\alpha } \eta _{\alpha }\int _{0}^{1} \left| \int _{\varOmega _r}\nabla \left( \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }} \left( {I}_{\alpha \beta }(X,s)\right) \right. \right. \nonumber \\&\quad \left. \left. \quad -\, \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }}\left( \tilde{I}_{\alpha \beta }(X,s) \right) \right) {{\mathrm{d}}}X\right| {{\mathrm{d}}}s\cdot \left| R_{\alpha \beta } \right| \nonumber \\&\quad \le \frac{4h}{V_r}\sum _{\alpha }\sum _{\beta \not =\alpha } \eta _{\alpha }\int _{0}^{1} \int _{\varOmega _r}\left| \nabla \left( \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }} \left( {I}_{\alpha \beta }(X,s)\right) \right. \right. \nonumber \\&\quad \left. \left. \quad -\, \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }}\left( \tilde{I}_{\alpha \beta }(X,s) \right) \right) \right| {{\mathrm{d}}}X{{\mathrm{d}}}s. \end{aligned}$$
(62)
Hence the error
$$\begin{aligned} \left| \int _{\varOmega _r}\left( \nabla \cdot {P}(X)- \nabla \cdot \tilde{P}(X)\right) {{\mathrm{d}}}X \right| \end{aligned}$$
(63)
is determined by
$$\begin{aligned} \int _{\varOmega _r}\left| \nabla \left( \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }} \left( {I}_{\alpha \beta }(X,s)\right) - \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }}\left( \tilde{I}_{\alpha \beta }(X,s) \right) \right) \right| {{\mathrm{d}}}X. \end{aligned}$$
(64)
Notice that
$$\begin{aligned}&\nabla \left( \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }} \left( {I}_{\alpha \beta }(X,s)\right) \right) = \frac{\partial }{\partial X_j}\frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}} \left( {I}_{\alpha \beta }(X,s)\right) e_ie_j \nonumber \\&\quad = \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} e_ie_j, \end{aligned}$$
(65)
where the Einstein summation convention is assumed for repeated indices i, j, k, and \(e_i (i=1,2,3)\) is the natural basis of \({\mathbb {R}}^3\).
In the same way
$$\begin{aligned}&\nabla \left( \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \right) \nonumber \\&\quad = \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} e_ie_j. \end{aligned}$$
(66)
Combining Eqs. (65) and (66) it follows that
$$\begin{aligned}&\left| \nabla \left( \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }} \left( {I}_{\alpha \beta }(X,s)\right) - \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }}\left( \tilde{I}_{\alpha \beta }(X,s) \right) \right) \right| \nonumber \\&\quad \le \sum _{i,j=1}^3 \left| \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right. \nonumber \\&\left. \qquad - \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right| , \end{aligned}$$
(67)
hence the expression (64) is determined by
$$\begin{aligned} ER_{ij}&:= \int _{\varOmega _r} \left| \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right. \nonumber \\&\left. \quad - \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j}\right| {{\mathrm{d}}}X\nonumber \\&= \left\| \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right. \nonumber \\&\left. \quad - \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{1,\varOmega _r}. \end{aligned}$$
(68)
Owing to
$$\begin{aligned}&\frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \nonumber \\&\quad \quad - \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \nonumber \\&\quad = \left[ \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) \right. \nonumber \\&\left. \qquad - \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \right] \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \nonumber \\&\qquad + \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \left[ \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} - \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right] , \end{aligned}$$
(69)
we have
Let us first estimate
$$\begin{aligned} \left\| \left[ \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) - \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \right] \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{1,\varOmega _r}. \end{aligned}$$
(71)
Due to the mean value theorem, there exists \(Y\in {\mathbb {R}}^3\) such that
$$\begin{aligned}&\frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) - \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \nonumber \\&\quad = \nabla _{r_{\alpha \beta }} \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}}(Y)\cdot \left( {I}_{\alpha \beta }(X,s)- \tilde{I}_{\alpha \beta }(X,s) \right) \nonumber \\&\quad = \frac{\partial ^3 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}\partial r_{\alpha \beta }^{(m)}}(Y) \left( {I}_{\alpha \beta }^{(m)}(X,s)- \tilde{I}_{\alpha \beta }^{(m)}(X,s) \right) , \end{aligned}$$
(72)
where Y can be written as
$$\begin{aligned} Y=\lambda I_{\alpha \beta }(X,s)+(1-\lambda )\tilde{I}_{\alpha \beta }(X,s)\quad (0\le \lambda \le 1). \end{aligned}$$
(73)
Let
$$\begin{aligned} M_3= \max _{\begin{array}{c} i,j,k=1,2,3 \\ \alpha \not =\beta ,Y\in {\mathbb {R}}^3 \end{array}} \left| \frac{\partial ^3 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(j)}\partial r_{\alpha \beta }^{(k)}}(Y) \right| , \end{aligned}$$
(74)
using Eq. (72) and Cauchy–Schwarz inequality we have
$$\begin{aligned}&\left\| \left[ \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) - \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \right] \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{1,\varOmega _r} \nonumber \\&\quad = \left\| \frac{\partial ^3 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}\partial r_{\alpha \beta }^{(m)}}(Y) \left( {I}_{\alpha \beta }^{(m)}(X,s)- \tilde{I}_{\alpha \beta }^{(m)}(X,s) \right) \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{1,\varOmega _r}\nonumber \\&\quad \le M_3 \sum _{m,k=1}^3 \left\| \left( {I}_{\alpha \beta }^{(m)}(X,s)- \tilde{I}_{\alpha \beta }^{(m)}(X,s) \right) \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{1,\varOmega _r}\nonumber \\&\quad \le M_3 \sum _{m=1}^3 \left\| {I}_{\alpha \beta }^{(m)}(X,s)- \tilde{I}_{\alpha \beta }^{(m)}(X,s) \right\| _{2,\varOmega _r} \sum _{k=1}^3 \left\| \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{2,\varOmega _r}. \end{aligned}$$
(75)
Moreover
$$\begin{aligned}&\left\| {I}_{\alpha \beta }^{(m)}(X,s)- \tilde{I}_{\alpha \beta }^{(m)}(X,s) \right\| _{2,\varOmega _r}\nonumber \\&\quad = \left\| \int _{0}^{1}F_{mn}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(n)}{{\mathrm{d}}}s'\right. \nonumber \\&\left. \qquad - \int _{0}^{1}\tilde{F}_{mn}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(n)}{{\mathrm{d}}}s' \right\| _{2,\varOmega _r} \nonumber \\&\quad = \left\| \int _{0}^{1} \left[ F_{mn}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(n)} \right. \right. \nonumber \\&\left. \left. \qquad - \tilde{F}_{mn}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(n)} \right] {{\mathrm{d}}}s' \right\| _{2,\varOmega _r}\nonumber \\&\quad \le \int _{0}^{1} \left\| F_{mn}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(n)} \right. \nonumber \\&\left. \qquad - \tilde{F}_{mn}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(n)} \right\| _{2,\varOmega _r} {{\mathrm{d}}}s'\nonumber \\&\quad \le 2h \sum _{n=1}^3\int _{0}^{1} \left\| F_{mn}\left( X+(s'-s)R_{\alpha \beta }\right) \right. \nonumber \\&\left. \qquad - \tilde{F}_{mn}\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r} {{\mathrm{d}}}s'\nonumber \\&\quad = 2h \sum _{n=1}^3\int _{0}^{1} \left\| \frac{\partial }{\partial X_n} R^{(m)}\left( X+(s'-s)R_{\alpha \beta }\right) \right. \nonumber \\&\left. \qquad - \frac{\partial }{\partial X_n} {\mathscr {I}}_hR^{(m)}\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r} {{\mathrm{d}}}s'. \end{aligned}$$
(76)
Notice that the interpolation operator \({\mathscr {I}}_h\) keeps first-degree polynomials unchanged, thanks to error analysis theory of finite element methods [2], we have
where C is a constant independent of h(we denote by C the irrelevant constants hereinafter). Thus
$$\begin{aligned}&\left\| {I}_{\alpha \beta }^{(m)}(X,s)- \tilde{I}_{\alpha \beta }^{(m)}(X,s) \right\| _{2,\varOmega _r} \nonumber \\&\quad \le Ch^2 \int _{0}^{1}\left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r}{{\mathrm{d}}}s'. \end{aligned}$$
(78)
Furthermore
$$\begin{aligned}&\left\| \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{2,\varOmega _r} \nonumber \\&\quad = \left\| \frac{\partial }{\partial X_j} \int _{0}^{1}F_{km}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(m)}{{\mathrm{d}}}s' \right\| _{2,\varOmega _r}\nonumber \\&\quad = \left\| \int _{0}^{1}\frac{\partial }{\partial X_j}F_{km}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(m)}{{\mathrm{d}}}s' \right\| _{2,\varOmega _r}\nonumber \\&\quad \le \int _{0}^{1}\left\| \frac{\partial }{\partial X_j}F_{km}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(m)} \right\| _{2,\varOmega _r}{{\mathrm{d}}}s'\nonumber \\&\quad \le 2h\int _{0}^{1}\left\| \frac{\partial }{\partial X_j}F_{km}\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r}{{\mathrm{d}}}s' \nonumber \\&\quad \le 2h\int _{0}^{1}\left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r}{{\mathrm{d}}}s'. \end{aligned}$$
(79)
Combining inequalities (75), (78) and (79) yields
Next we estimate
$$\begin{aligned} \left\| \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \left[ \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} - \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right] \right\| _{1,\varOmega _r}. \end{aligned}$$
(81)
Let
$$\begin{aligned} M_2= \max _{\begin{array}{c} i,j=1,2,3 \\ \alpha \not =\beta , Y\in {\mathbb {R}}^3 \end{array}} \left| \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(j)}}(Y) \right| , \end{aligned}$$
(82)
then
$$\begin{aligned}&\left\| \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \left[ \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} - \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right] \right\| _{1,\varOmega _r}\nonumber \\&\quad \le M_2\sum _{k=1}^3 \left\| \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} - \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{1,\varOmega _r}. \end{aligned}$$
(83)
Moreover
$$\begin{aligned}&\left\| \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} - \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{1,\varOmega _r} \nonumber \\&\quad = \left\| \frac{\partial }{\partial X_j} \int _{0}^{1}F_{km}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(m)}{{\mathrm{d}}}s' \right. \nonumber \\&\left. \qquad - \frac{\partial }{\partial X_j} \int _{0}^{1}\tilde{F}_{km}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(m)}{{\mathrm{d}}}s' \right\| _{1,\varOmega _r}\nonumber \\&\quad = \left\| \int _{0}^{1} \left[ \frac{\partial }{\partial X_j} F_{km}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(m)} \right. \right. \nonumber \\&\left. \left. \qquad - \frac{\partial }{\partial X_j} \tilde{F}_{km}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(m)} \right] {{\mathrm{d}}}s' \right\| _{1,\varOmega _r}\nonumber \\&\quad \le \int _{0}^{1} \left\| \frac{\partial }{\partial X_j} F_{km}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(m)} \right. \nonumber \\&\left. \qquad - \frac{\partial }{\partial X_j} \tilde{F}_{km}\left( X+(s'-s)R_{\alpha \beta }\right) R_{\alpha \beta }^{(m)} \right\| _{1,\varOmega _r} {{\mathrm{d}}}s'\nonumber \\&\quad \le 2h \sum _{m=1}^3 \int _{0}^{1} \left\| \frac{\partial }{\partial X_j} F_{km}\left( X+(s'-s)R_{\alpha \beta }\right) \right. \nonumber \\&\left. \qquad - \frac{\partial }{\partial X_j} \tilde{F}_{km}\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{1,\varOmega _r} {{\mathrm{d}}}s'\nonumber \\&\quad = 2h \sum _{m=1}^3\int _{0}^{1} \left\| \frac{\partial ^2}{\partial X_j\partial X_m} R^{(k)}\left( X+(s'-s)R_{\alpha \beta }\right) \right. \nonumber \\&\left. \qquad - \frac{\partial ^2}{\partial X_j\partial X_m} {\mathscr {I}}_hR^{(k)}\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{1,\varOmega _r} {{\mathrm{d}}}s'. \end{aligned}$$
(84)
Again, using error analysis theory of finite element methods it follows that
$$\begin{aligned}&\left\| \frac{\partial ^2}{\partial X_j\partial X_m} R^{(k)}\left( X+(s'-s)R_{\alpha \beta }\right) \right. \nonumber \\&\left. \qquad - \frac{\partial ^2}{\partial X_j\partial X_m} {\mathscr {I}}_hR^{(k)}\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{1,\varOmega _r} \nonumber \\&\quad \le Ch^{3/2} \left\| \nabla ^2 R\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r}\nonumber \\&\quad = Ch^{3/2} \left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r}. \end{aligned}$$
(85)
Combining inequalities (83) through (85) we have
$$\begin{aligned}&\left\| \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \left[ \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} - \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right] \right\| _{1,\varOmega _r} \nonumber \\&\quad \le Ch^{5/2}M_2 \int _{0}^{1} \left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r} {{\mathrm{d}}}s'. \end{aligned}$$
(86)
Now combining inequalities (70), (80) and (86) leads to
$$\begin{aligned} ER_{ij}&= \left\| \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( {I}_{\alpha \beta }(X,s)\right) \frac{\partial {I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right. \nonumber \\&\left. \qquad -\, \frac{\partial ^2 E_{\alpha }}{\partial r_{\alpha \beta }^{(i)}\partial r_{\alpha \beta }^{(k)}} \left( \tilde{I}_{\alpha \beta }(X,s)\right) \frac{\partial \tilde{I}_{\alpha \beta }^{(k)}(X,s)}{\partial X_j} \right\| _{1,\varOmega _r} \nonumber \\&\quad \le Ch^3M_3 \left( \int _{0}^{1}\left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r}{{\mathrm{d}}}s' \right) ^2 \nonumber \\&\qquad + Ch^{5/2}M_2 \int _{0}^{1} \left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r} {{\mathrm{d}}}s'. \end{aligned}$$
(87)
Furthermore, using inequality (67) we have
$$\begin{aligned}&\int _{\varOmega _r}\left| \nabla \left( \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }} \left( {I}_{\alpha \beta }(X,s)\right) - \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }}\left( \tilde{I}_{\alpha \beta }(X,s) \right) \right) \right| {{\mathrm{d}}}X \nonumber \\&\quad \le Ch^3M_3 \left( \int _{0}^{1}\left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r}{{\mathrm{d}}}s' \right) ^2 \nonumber \\&\qquad + Ch^{5/2}M_2 \int _{0}^{1} \left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r} {{\mathrm{d}}}s'. \end{aligned}$$
(88)
Combining inequalities (62) and (88) we finally derive
$$\begin{aligned}&\left| \int _{\varOmega _r}\left( \nabla \cdot {P}(X)- \nabla \cdot \tilde{P}(X)\right) {{\mathrm{d}}}X \right| \nonumber \\&\quad \le \frac{1}{2h^2}\sum _{\alpha }\sum _{\beta \not =\alpha } \eta _{\alpha }\int _{0}^{1} \int _{\varOmega _r}\left| \nabla \left( \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }} \left( {I}_{\alpha \beta }(X,s)\right) \right. \right. \nonumber \\&\left. \left. \qquad -\, \frac{\partial E_{\alpha }}{\partial r_{\alpha \beta }}\left( \tilde{I}_{\alpha \beta }(X,s) \right) \right) \right| {{\mathrm{d}}}X{{\mathrm{d}}}s \nonumber \\&\quad \le ChM_3 \sum _{\alpha }\sum _{\beta \not =\alpha }\int _{0}^{1} \left( \int _{0}^{1}\left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r}{{\mathrm{d}}}s' \right) ^2 {{\mathrm{d}}}s \nonumber \\&\qquad + Ch^{1/2}M_2 \sum _{\alpha }\sum _{\beta \not =\alpha }\int _{0}^{1} \int _{0}^{1}\left\| \nabla F\left( X+(s'-s)R_{\alpha \beta }\right) \right\| _{2,\varOmega _r}{{\mathrm{d}}}s' {{\mathrm{d}}}s. \end{aligned}$$
(89)
