Isogeometric Galerkin discretization

By [F1], $E_{0}z\in H^1({\mathbb{R}}^d)$ and $E_{0}z=z$ on $U$, $E_{0}z=0$ off $U$. By [F2], choose $q_n\in C^{\mathrm{\infty }}({\mathbb{R}}^d)\cap H^1({\mathbb{R}}^d)$ with $q_n\to E_{0}z$ in $H^1({\mathbb{R}}^d)$. Choose $\chi$ from [F3].

Multiplication by $\chi$ is bounded on $H^1({\mathbb{R}}^d)$. Indeed, for a weak derivative and a test function $\psi$, the weak identity gives $\int \chi q{\mathrm{\partial }}_j\psi =\int q{\mathrm{\partial }}_j(\chi \psi )-\int q({\mathrm{\partial }}_j\chi )\psi =-\int (\chi {\mathrm{\partial }}_{j}q+q{\mathrm{\partial }}_j\chi )\psi ,$ so $D_j(\chi q)=\chi D_{j}q+q{\mathrm{\partial }}_j\chi$; the boundedness of $\chi$ and its derivatives gives the $H^1$ norm bound. Since $\chi E_{0}z=z$ almost everywhere, $\chi q_n\to z$ in $H^1(U)$. Every $\chi q_n$ is smooth and compactly supported in $U$, so [F4] proves $z\in H_0^1(U)$.

Fix a cell $K$ and one vertex $x_0$. Put $\mathrm{\ell }(x)=g(x_0)+\mathrm{\nabla }g(x_0)\cdot (x-x_0)$ and $r=g-\mathrm{\ell }$. The cell diameter is $\sqrt{d}h$, so [F2] gives $\mathrm{\mid }r(x)\mathrm{\mid }\le (d\mathrm{/}2)h^2{M}_g$ on $K$ and $\mathrm{\mid }\mathrm{\nabla }r(x)\mathrm{\mid }\le \sqrt{d}h{M}_g$. The tensor Q1 interpolant reproduces $\mathrm{\ell }$, hence $g-I_{K}g=r-I_{K}r$.

Let ${\varphi }_{\nu }$ be the $2^d$ tensor nodal shape functions. They satisfy ${\varphi }_{\nu }\ge 0$, ${\sum }_{\nu }{\varphi }_{\nu }=1$, and $\mathrm{\mid }\mathrm{\nabla }{\varphi }_{\nu }\mathrm{\mid }\le \sqrt{d}\mathrm{/}h$. Thus $\mathrm{\mid }{I}_{K}r\mathrm{\mid }\le (d\mathrm{/}2)h^2{M}_g$ and $\mathrm{\mid }\mathrm{\nabla }{I}_{K}r\mathrm{\mid }\le {\sum }_{\nu }\mathrm{\mid }r(x_{\nu })\mathrm{\mid }\mathrm{\mid }\mathrm{\nabla }{\varphi }_{\nu }\mathrm{\mid }\le 2^{d-1}d\sqrt{d}h{M}_g\mathrm{.}$ Combining these with step 1.1 yields $\mathrm{\mid }g-I_{K}g\mathrm{\mid }\le d{h}^2{M}_g$ and $\mathrm{\mid }\mathrm{\nabla }(g-I_{K}g)\mathrm{\mid }\le (\sqrt{d}+2^{d-1}d\sqrt{d})h{M}_g$.

Adjacent cells use the same vertex values, so their local interpolants agree on their common faces; consequently $I_{h}g$ is continuous and belongs to the tensor degree-one spline space defined by [F1]. Square the pointwise bounds of step 2.1, integrate on each cell, and sum over the unit-volume partition. The resulting global $L^2$ errors are at most $d{h}^2{M}_g$ and $(\sqrt{d}+2^{d-1}d\sqrt{d})h{M}_g$ for value and gradient. Since $h\le 1$, the Euclidean combination defining the $H^1$ norm is at most their sum, which is bounded by $C_{d}h{M}_g$.

First let $v\in C^{\mathrm{\infty }}(\mathrm{\Omega })\cap H^1(\mathrm{\Omega })$. On each open knot cell the differentiability expansions of $F$ and $v$ give ${\mathrm{\nabla }}_{\xi }(v\circ F)=D{F}^{\mathsf{T}}({\mathrm{\nabla }}_{x}v)\circ F$. Fix a test function compactly supported in $\widehat{\mathrm{\Omega }}$. Its support stays away from the exterior patch boundary. On every part of a cell where the test function is nonzero, its image under $F$ lies in a compact subset of $\mathrm{\Omega }$, so $v\circ F$ and its first derivatives are bounded there; near exterior faces the test function vanishes. Apply the divergence theorem [F6] to $(v\circ F)\varphi e_j$ cell by cell. The values of $v\circ F$ and $\varphi$ agree on either side of every shared face, so opposite normal fluxes cancel, and exterior terms vanish. This proves the weak-derivative identity without requiring $v$ to extend to $\mathrm{\partial }\mathrm{\Omega }$. Cellwise change of variables [F2], summed over the finite partition using finite additivity of the integral, yields $\mathrm{\parallel }v\circ F{\mathrm{\parallel }}_{L^2(\widehat{\mathrm{\Omega }})}^2\le j_{-}^{-1}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2,\mathrm{\parallel }{\mathrm{\nabla }}_{\xi }(v\circ F){\mathrm{\parallel }}_{L^2(\widehat{\mathrm{\Omega }})}^2\le L^2{j}_{-}^{-1}\mathrm{\parallel }{\mathrm{\nabla }}_{x}v{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\mathrm{.}$ Thus $\mathrm{\parallel }v\circ F{\mathrm{\parallel }}_{H^1(\widehat{\mathrm{\Omega }})}\le j_{-}^{-1\mathrm{/}2}\sqrt{1+L^2}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1(\mathrm{\Omega })}$. The same argument applied to $F^{-1}$, whose Jacobian is $J_F^{-1}$ and derivative is bounded by $L$, gives the reverse estimate.

For arbitrary $u\in H^1(\mathrm{\Omega })$, choose $v_n\in C^{\mathrm{\infty }}(\mathrm{\Omega })\cap H^1(\mathrm{\Omega })$ with $v_n\to u$ in $H^1$ by [F1]. Step 1.1 applied to $v_n-v_m$ makes $v_n\circ F$ Cauchy in $H^1(\widehat{\mathrm{\Omega }})$; completeness [F4] gives a limit $\widehat{u}\in H^1(\widehat{\mathrm{\Omega }})$. The same change-of-variables estimate in step 1.1 implies $v_n\circ F\to u\circ F$ in $L^2(\widehat{\mathrm{\Omega }})$, so this limit is the composition class. The gradient formula follows by taking the $L^2$ limit of the step 1.1 gradients: the bound on $DF$ and the change-of-variables estimate imply $D{F}^{\mathsf{T}}(\mathrm{\nabla }{v}_n)\circ F\to D{F}^{\mathsf{T}}(\mathrm{\nabla }u)\circ F$ in $L^2$. Repeating the argument with $F^{-1}$ proves the reverse estimate and inverse pullback.

Suppose $u\in H_0^1(\mathrm{\Omega })$ and choose ${\phi }_n\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ converging to $u$ in $H^1$ by [F4]. Each ${\phi }_n\circ F$ belongs to $H^1(\widehat{\mathrm{\Omega }})$ by step 1.1 and has compact support in $\widehat{\mathrm{\Omega }}$, because $F^{-1}$ maps compact subsets of $\mathrm{\Omega }$ to compact subsets of $\widehat{\mathrm{\Omega }}$. Applying [F5] gives ${\phi }_n\circ F\in H_0^1(\widehat{\mathrm{\Omega }})$. The forward estimate in step 2.1 gives convergence in $H^1$, and closedness in [F4] implies $u\circ F\in H_0^1(\widehat{\mathrm{\Omega }})$. Applying the same reasoning to $F^{-1}$ proves the converse. Finally, the scalar energy identity follows by substituting ${\mathrm{\nabla }}_{x}u\circ F=D{F}^{-\mathsf{T}}{\mathrm{\nabla }}_{\xi }\widehat{u}$ and $dx=J_{F}d\xi$ into the defining integral. For a vector field with row-gradient convention, ${\mathrm{\nabla }}_{x}v=({\mathrm{\nabla }}_{\xi }\widehat{v})D{F}^{-1}$; hence ${\epsilon }_x(v)\circ F=\mathrm{sym}(({\mathrm{\nabla }}_{\xi }\widehat{v})D{F}^{-1})$ and ${\mathrm{div}}_{x}v\circ F=\mathrm{tr}(({\mathrm{\nabla }}_{\xi }\widehat{v})D{F}^{-1})$, which gives the stated elasticity transformation.

Fix a component $q\in \{1,\dots ,m\}$, a test function $\varphi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and a coordinate $j$. The finite physical partition and face-equality convention are those of [F1]. On each cell $K$, the vector field $G_K:=v_q\varphi e_j$ is $C^1$ up to the cell boundary. By [F3], ${\int }_K{v}_q{\mathrm{\partial }}_j\varphi +{\mathrm{\partial }}_j{v}_q\varphi ={\int }_{\mathrm{\partial }K}{v}_q\varphi n_{K,j}dS\mathrm{.}$ Summing this identity over the finitely many cells is legitimate because both volume terms are integrable by the hypotheses and [F4]. On a shared face the two normal fluxes add to $(v_{q,K}-v_{q,K^{\mathrm{\prime }}})\varphi n_{K,j}$, which is zero by the matching-trace assumption; edge overlaps have surface measure zero by [F2]. The exterior boundary term is zero because $\varphi$ is compactly supported in $\mathrm{\Omega }$. Therefore ${\int }_{\mathrm{\Omega }}{v}_q{\mathrm{\partial }}_j\varphi =-{\sum }_K{\int }_K{\mathrm{\partial }}_j{v}_q\varphi \mathrm{.}$

For every component $q$ and coordinate $j$, define $g_{qj}$ to equal the classical ${\mathrm{\partial }}_j{v}_q$ on each open cell and arbitrarily on the cell faces. The faces have $d$-dimensional measure zero under the finite piecewise $C^1$ presentation, and the finite sum of cellwise $L^2$ norms is finite, so $g_{qj}\in L^2(\mathrm{\Omega })$. Step 1.1 gives the weak-derivative identity for every test function, hence $D_j{v}_q=g_{qj}$ weakly. Each component of $v$ is also square integrable on the finite bounded cell partition; [F4] therefore gives $v\in H^1(\mathrm{\Omega };{\mathbb{R}}^m)$ with the asserted componentwise gradient.

Each tensor product $N_{\alpha }$ is nonnegative by [F1]. On any open tensor knot cell, each coordinate lies in one nonempty knot interval, so finite distributivity and [F1] give ${\sum }_{\alpha \in \mathcal{I}}{N}_{\alpha }(\xi )={\prod }_{r=1}^d({\sum }_{i_r}{B}_{i_r,p_r,{\mathrm{\Xi }}_r}^{(r)}({\xi }_r))=1.$ Hence $0\le N_{\alpha }\le 1$ on the cell. The endpoint values follow by the one-sided endpoint convention in B-spline spaces and knot conventions.

Since every coefficient $w_{\alpha }$ is between $w_{-}$ and $w_{+}$, the identity in step 1.1 yields $w_{-}{\sum }_{\alpha }{N}_{\alpha }\le W={\sum }_{\alpha }{w}_{\alpha }{N}_{\alpha }\le w_{+}{\sum }_{\alpha }{N}_{\alpha },$ so $w_{-}\le W\le w_{+}$. Thus each $R_{\alpha }$ is well-defined and nonnegative, and ${\sum }_{\alpha }{R}_{\alpha }=\frac{{\sum }_{\alpha }{w}_{\alpha }{N}_{\alpha }}{W}=1.$ For any coefficients $c_{\alpha }$, ${\sum }_{\alpha }{c}_{\alpha }{R}_{\alpha }=(1\mathrm{/}W){\sum }_{\alpha }(c_{\alpha }{w}_{\alpha })N_{\alpha }$. Multiplication of each coefficient by the nonzero $w_{\alpha }$ is a bijection on the finite coefficient space, so the span equals $S_h\mathrm{/}W$.

On the open knot cell all $N_{\alpha }$ and $W$ are polynomial and $W\ge w_{-}>0$ by step 2.1. The ordinary quotient rule gives the displayed gradient formula. Also $\mathrm{\parallel }\mathrm{\nabla }W{\mathrm{\parallel }}_{\mathrm{\infty }}\le {\sum }_{\alpha }{w}_{\alpha }\mathrm{\parallel }\mathrm{\nabla }{N}_{\alpha }{\mathrm{\parallel }}_{\mathrm{\infty }}\le G_W$ and $N_{\alpha }\le 1$ by step 1.1. Applying these bounds to the quotient rule proves the stated derivative estimate. The finite number of polynomial pieces makes each $G_N$ finite.

By [F2], $K_F$ and $K_G$ are the unit-Laplace pullback tensors. Fix a point where the hypotheses hold and write $A=DF$, $B=DG$; [F1] supplies the cellwise map bounds. For any square matrix with columns $c_1,\dots ,c_d$, if they are dependent its determinant is zero. If independent, Gram--Schmidt replaces each $c_i$ by $q_i=c_i-{\sum }_{k<i}{\mathrm{proj}}_{q_k}{c}_i$; subtracting combinations of earlier columns preserves the determinant, the $q_i$ are orthogonal, and $\mathrm{\parallel }{q}_i\mathrm{\parallel }\le \mathrm{\parallel }{c}_i\mathrm{\parallel }$ by the Pythagorean identity. Since $\mathrm{\mid }\det [q_1\cdots q_d]{\mathrm{\mid }}^2=\det (Q^{\mathsf{T}}Q)={\prod }_i\mathrm{\parallel }{q}_i{\mathrm{\parallel }}^2$, this proves $\mathrm{\mid }\det [c_1\cdots c_d]\mathrm{\mid }\le {\prod }_i\mathrm{\parallel }{c}_i\mathrm{\parallel }$. Now multilinearity replaces the columns of $B$ by those of $A$ one at a time. Each determinant difference has one column of norm at most $\mathrm{\parallel }A-B{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \delta$ and $d-1$ columns of norm at most $M$, so $\mathrm{\mid }\det A-\det B\mathrm{\mid }\le d{M}^{d-1}\delta$. The same volume bound gives $\mathrm{\mid }\det B\mathrm{\mid }\le M^d$.

The inverse identity $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$ gives $\mathrm{\parallel }{A}^{-1}-B^{-1}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le M^2\delta$. Put $S_A=A^{-1}{A}^{-\mathsf{T}}$ and $S_B=B^{-1}{B}^{-\mathsf{T}}$. Add and subtract $B^{-1}{A}^{-\mathsf{T}}$ to obtain $\mathrm{\parallel }{S}_A-S_B{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \mathrm{\parallel }{A}^{-1}-B^{-1}\mathrm{\parallel }\mathrm{\parallel }{A}^{-1}\mathrm{\parallel }+\mathrm{\parallel }{B}^{-1}\mathrm{\parallel }\mathrm{\parallel }{A}^{-1}-B^{-1}\mathrm{\parallel }\le 2M^3\delta ,$ while $\mathrm{\parallel }{S}_A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le M^2$.

Write $(\det A)S_A-(\det B)S_B=(\det A-\det B)S_A+(\det B)(S_A-S_B)$. The bounds in steps 1.1 and 2.1 give $\mathrm{\parallel }{K}_F-K_G{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le d{M}^{d-1}\delta M^2+M^{d}2M^3\delta =(d{M}^{d+1}+2M^{d+3})\delta \mathrm{.}$ This proves the pointwise tensor estimate almost everywhere.

Apply the pointwise estimate to the integrand and then the integral Cauchy--Schwarz inequality [F3] on the unit box: $\mid \int (K_F-K_G)\mathrm{\nabla }v\cdot \mathrm{\nabla }w\mid \le C\delta \int \mathrm{\mid }\mathrm{\nabla }v\mathrm{\mid }\mathrm{\mid }\mathrm{\nabla }w\mathrm{\mid }\le C\delta \mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2,$ where $C=d{M}^{d+1}+2M^{d+3}$. The integral is well-defined because the tensor difference is essentially bounded and the gradients are square integrable.

Work on the finite common refinement and set $e:=g-s$. The continuous positive denominator is polynomial on each geometry cell, so $g$, $s$, $e$, $s\mathrm{/}W$ and $e\mathrm{/}W$ are continuous and C1 up to each refined cell boundary. Their classical cell gradients are square integrable on this finite partition; [F3] therefore puts them in H1 with these weak gradients. On each cell, $u-v_h=(\widehat{u}-s\mathrm{/}W)\circ F^{-1}=(e\mathrm{/}W)\circ F^{-1}$. Since $W\ge w_{-}>0$, the cellwise quotient rule gives $\mid \frac{e}{W}\mid \le \frac{\mathrm{\mid }e\mathrm{\mid }}{w_{-}},\mid \mathrm{\nabla }\text{\hspace{0.17em}⁣}\left(\frac{e}{W}\right)\mid \le \frac{\mathrm{\mid }\mathrm{\nabla }e\mathrm{\mid }}{w_{-}}+\frac{G_W\mathrm{\mid }e\mathrm{\mid }}{w_{-}^2}\mathrm{.}$ Taking the $H^1$ norm and using the triangle inequality yields $\mathrm{\parallel }e\mathrm{/}W{\mathrm{\parallel }}_{H^1(\widehat{\mathrm{\Omega }})}\le C_W\mathrm{\parallel }e{\mathrm{\parallel }}_{H^1(\widehat{\mathrm{\Omega }})}$.

The traces of $e\mathrm{/}W$ match because the traces of $\widehat{u}$, $s$, and $W$ match and $W$ is positive. Its cell gradients are square integrable by step 1.1, so [F3] gives $e\mathrm{/}W\in H^1(\widehat{\mathrm{\Omega }})$. By [A1], [F2] maps $s\mathrm{/}W\in H^1(\widehat{\mathrm{\Omega }})$ to $v_h\in H^1(\mathrm{\Omega })$. Applying its inverse-map estimate to $(e\mathrm{/}W)\circ F^{-1}$ gives $\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_{H^1(\mathrm{\Omega })}\le C_F\mathrm{\parallel }e\mathrm{/}W{\mathrm{\parallel }}_{H^1(\widehat{\mathrm{\Omega }})}\le C_F{C}_W\mathrm{\parallel }g-s{\mathrm{\parallel }}_{H^1(\widehat{\mathrm{\Omega }})}\mathrm{.}$ This is the claimed estimate, with all constants retaining the map and weight dependence.

In the normalized cube every point has some ordering of its $d$ coordinates, hence belongs to at least one displayed simplex; strict order specifies exactly one interior. For a permutation $\pi$, the closed ordered region is the convex hull of the chain vertices $0,e_{\pi (d)},e_{\pi (d)}+e_{\pi (d-1)},\dots ,{\sum }_{r=1}^d{e}_{\pi (r)}$: the barycentric coefficients are $1-t_{\pi (d)}$, $t_{\pi (d)}-t_{\pi (d-1)},\dots ,t_{\pi (2)}-t_{\pi (1)},t_{\pi (1)}$. Thus a tie between consecutive coordinates sets exactly the corresponding barycentric coefficient to zero. For two permutations, the common weak order groups precisely the coordinates whose order is reversed into equal blocks; their intersection is the convex hull of the common chain vertices at the block boundaries, hence is a whole face of each simplex. This proves the subdivision is a simplicial complex with disjoint interiors. On a rectangular face one coordinate is $0$ or $1$; the remaining subdivision orders the remaining normalized coordinates. Adjacent rectangles have the same intervals in those coordinates and the same increasing-axis convention, so their induced face subdivisions coincide. The affine diagonal maps preserve these face identities. Therefore the finite subdivision is a matching affine simplicial mesh as required in [F2].

On a closed rectangle each map is a quotient of finite polynomials with denominator bounded below by a positive number. Its cell expression therefore extends smoothly across the rectangle into a neighborhood. Its derivatives, including second derivatives, are bounded there after shrinking the neighborhood. The determinant extends continuously with a positive lower bound on the closed cell. Its restriction to each closed simplex is injective by global injectivity of the map on $\bar{Q}$. This restriction actually extends injectively to some neighborhood of that simplex: otherwise there exist distinct pairs of points at distance at most $1\mathrm{/}n$ from the simplex with equal images. Compactness [F3] gives convergent subsequences with limits in the simplex; equality of images and injectivity force the same limit, contradicting local injectivity [F3] near that limit. A smaller neighborhood retains positive determinant, so the restriction is a neighborhood diffeomorphism. These arguments apply separately to $F$ and $G$.

Continuity of the original maps makes the restricted traces agree on shared simplex facets, including those lying on original knot planes. Global injectivity preserves disjoint interiors and makes intersections of physical cells precisely the images of the corresponding reference intersections. Their union is the original image. The map, inverse, determinant and finite second-derivative bounds from step 1.2 give all curved-cell requirements [F2]. The reference simplices here are actual parameter simplices; if one instead precomposes with a map from a unit simplex, its Jacobian scaling factors must be retained.

The cell restrictions do not alter the global maps, their physical images, or their value and almost-everywhere derivative differences. Every introduced reference-simplex facet lies in an affine hyperplane, the image of an affine $C^1$ map from ${\mathbb{R}}^{d-1}$ into ${\mathbb{R}}^d$. By [F4] and [A1] each such hyperplane is null, and the finite union of introduced facets is null. Thus taking essential suprema over the refined interiors gives the same $e_1$; continuity gives the same $e_0$. To state separate finite error bounds as required by [F2], write $P_{\alpha }^F,P_{\beta }^G$ for the two finite control-point lists and $L_F,L_G$ for their admissibility derivative bounds. The nonnegative rational coefficients in [F1] sum to one, so $e_0\le {\max }_{\alpha }\mathrm{\mid }{P}_{\alpha }^F\mathrm{\mid }+{\max }_{\beta }\mathrm{\mid }{P}_{\beta }^G\mathrm{\mid }$; the operator-norm triangle inequality gives $e_1\le L_F+L_G$. The two positive determinant lower bounds are their respective admissibility constants $j_{-}^F,j_{-}^G$. These coarse bounds establish the finite geometry interface; no smallness or mesh-dependent convergence follows from admissibility alone. Since both maps are bi-Lipschitz bijections of the closed box, $G\circ F^{-1}$ is the required homeomorphism between the closed image domains. The within-mesh trace matching in step 2.1 supplies boundary-trace compatibility in [F2]; no exterior trace equality between distinct maps has been proved.

Each normalized ordered-coordinate simplex belongs to a finite set depending only on $d$. Under diagonal scaling, diameter increases by at most ${\max }_r{\mathrm{\ell }}_r$ and inradius decreases by at most the factor ${\min }_r{\mathrm{\ell }}_r$ relative to the normalized simplex. Hence the diameter/inradius ratio is at most $\kappa$ times the largest ratio in that fixed finite set. This proves the stated reference shape bound. The remaining uniform curved-map bounds are exactly the additional displayed family assumptions; finite-grid smoothness in step 1.2 supplies only finite-grid constants. No analysis-space change was made in the subdivision.