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Multipatch approximation of the de Rham sequence and its traces in isogeometric analysis

Abstract

We define a conforming B-spline discretisation of the de Rham complex on multipatch geometries. We introduce and analyse the properties of interpolation operators onto these spaces which commute w.r.t. the surface differential operators. Using these results as a basis, we derive new convergence results of optimal order w.r.t. the respective energy spaces and provide approximation properties of the spline discretisations of trace spaces for application in the theory of isogeometric boundary element methods. Our analysis allows for a straight forward generalisation to finite element methods.

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Abbreviations

\(H^s(\varOmega )\) :

For \(s>0\), standard Sobolev spaces, for \(s<0\) corresponding dual space

\(H^s(\mathrm d,\varOmega )\) :

Sobolev spaces w.r.t. a graph norm induced by \(\mathrm { d}\)

\(\pmb {H}_\times ^s\) :

Vectorial trace spaces

\(\pmb {H}_\times ^{s}({\text {div}}_\varGamma ,\varGamma )\) :

Trace of \(\pmb {H}^s({{\,\mathrm{{curl}}\,}},\varOmega )\) w.r.t. \(\pmb {\gamma }_t\)

\(H_\text {pw}^s(\varGamma )\) :

Sobolev space of patchwise regularity

\(\pmb {H}_\text {pw}^s({\text {div}}_\varGamma ,\varGamma )\) :

\(\pmb {H}^{0}({\text {div}}_\varGamma ,\varGamma )\) with a patchwise regularity of \({\pmb {H}}^{s}({\text {div}}_\varGamma ,\varGamma _j)\)

\(\pmb {H}_*^s({\text {div}}_\varGamma ,\varGamma )\) :

\(\pmb {H}_\times ^{-1/2}({\text {div}}_\varGamma ,\varGamma )\) with a patchwise regularity of \({\pmb {H}}^{s}({\text {div}}_\varGamma ,\varGamma _j)\)

\(K^s_j\) :

Kernel of \(\gamma _{\pmb {n},j}\) w.r.t. \(\pmb {H}^s({\text {div}},\varGamma _j)\)

\(K^{\mathbb {S}}_j\) :

Kernel of \(\gamma _{\pmb {n},j}\) w.r.t. \(\pmb {{\mathbb {S}}}^1_{\pmb {p},\pmb {\varXi }}(\varGamma )\)

\((K^s_j)'\) :

Dual of \(K^s_j\) w.r.t. \(\pmb {H}^0({\text {div}}_\varGamma ,\varGamma )\)

p :

Polynomial degree

\(\pmb {p}\) :

List of pairs of polynomial degrees

\(\varXi \) :

Knot vector

\(\pmb {\varXi }\) :

List of pairs of knot vectors

\(b_i^p\) :

B-spline basis function

\(S_p(\varXi )\) :

B-spline space of degree p over knot vector \(\varXi \)

\(S_{p_1,\ldots ,p_\ell }(\varXi _1,\ldots ,\varXi _\ell )\) :

Tensor product B-spline space of degrees \(p_1,\ldots ,p_\ell \) over knot vectors \(\varXi _1,\ldots ,\varXi _\ell \)

\({\tilde{I}}\),\(\tilde{Q}\) :

Support extension of IQ

\({\mathbb {S}}^0_{\pmb {p},\pmb {\varXi }}(\varGamma )\) :

\(H^1(\varGamma )\) conforming spline space in the physical domain on a multipatch geometry \(\varGamma \)

\(\pmb {{\mathbb {S}}}_{\pmb {p},\pmb {\varXi }}^1(\varGamma )\) :

\(\pmb {H}({\text {div}}_\varGamma ,\varGamma )\) conforming spline space in the physical domain on a multipatch geometry \(\varGamma \)

\({\mathbb {S}}^2_{\pmb {p},\pmb {\varXi }}(\varGamma )\) :

\(L^2(\varGamma )\) conforming spline space in the physical domain on a multipatch geometry \(\varGamma \)

\(\tilde{\varPi }_{p,\varXi }\) :

One-dimensional multipatch quasi-interpolation operator

\(\tilde{\varPi }^\partial _{p,\varXi }\) :

Commuting one-dimensional multipatch quasi-interpolation operator

\(\tilde{\varPi }_{\pmb {p},\pmb {\varXi }}^0\) :

\(H^1((0,1)^2)\) conforming, commuting multipatch quasi-interpolation operator for the reference domain

\(\tilde{\pmb {\varPi }}_{\pmb {p},\pmb {\varXi }}^1\) :

\(\pmb {H}({\text {div}},(0,1)^2)\) conforming, commuting multipatch quasi-interpolation operator for the reference domain

\(\tilde{\varPi }_{\pmb {p},\pmb {\varXi }}^2\) :

\(L^2((0,1)^2)\) conforming, commuting multipatch quasi-interpolation operator for the reference domain

\({{\tilde{\varPi }}^0_\varGamma }\) :

\(H^1(\varGamma )\) conforming, commuting multipatch quasi-interpolation operator for the physical domain

\({\pmb {\tilde{\varPi }}^1_\varGamma }\) :

\(\pmb {H}({\text {div}},\varGamma )\) conforming, commuting multipatch quasi-interpolation operator for the physical domain

\({{\tilde{\varPi }}^2_\varGamma }\) :

\(L^2(\varGamma )\) conforming, commuting multipatch quasi-interpolation operator for the physical domain

\(\pi \), \(\pi _j\) :

Quasi-optimal \(\pmb {H}({\text {div}}_\varGamma ,\varGamma )\) projection, defined patchwise via \(\pi _j\)

\(\mathscr {P}_s\) :

Orthogonal projection with respect to the \(H^s(\varGamma )\) scalar product

\(\mathscr {P}_{{\text {div}}}\) :

Orthogonal projection with respect to the \(\pmb {H}^0({\text {div}}_\varGamma ,\varGamma )\) scalar product

\(\mathscr {P}_\times \) :

Orthogonal projection with respect to the \(\pmb {H}^{-1/2}_\times ({\text {div}}_\varGamma ,\varGamma )\) scalar product

\(\pmb {F}_j\) :

Geometry mapping

\(\iota _*(\pmb {F})\) :

For \(* = 0,1,2,\) pull-backs induced by a given diffeomorphism \(\pmb {F}\)

\(\kappa (\pmb {x})\) :

Surface measure at a given point \(\pmb {x}\)

\(\pmb {n}_{\pmb {x}}\) :

Outer unit normal at \(\pmb {x}\)

\(\gamma _0\) :

Dirichlet trace

\(\pmb {\gamma }_t\) :

Rotated tangential trace operator

\(\pmb {\gamma }_0\) :

Vectorial Dirichlet trace operator

\(\gamma _{\pmb {n}}\) :

Normal trace operator

\(\gamma _{\pmb {n},j}\) :

Inter-patch normal trace on \(\partial \varGamma _j\)

\({\mathbb {K}}\) :

Either \(\mathbb R\) or \(\mathbb C\)

\(\lesssim \) :

\(\le \) up to a constant independent of h

\(\simeq \) :

\(=\) up to a constant independent of h

h :

Mesh size

i :

Index referring to a specific basis function, knot or mesh element

j :

Index referring to a specific patch

k :

Dimension of a discrete space on [0, 1]

\(\ell \) :

Spatial dimension

s, t, r :

Index referring to the regularity of a Sobolev space

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Acknowledgements

The authors want to express their sincere gratitude to Jacopo Corno, who provided feedback on early versions of the draft. Moreover, we want to thank the anonymous referee, whose review was of exceptional quality and helped us to improve the manuscript significantly. The work of A. Buffa and R. Vázquez was partially supported by the European Research Council through the H2020 ERC Advanced Grant no. 694515 CHANGE. J. Dölz is an Early Postdoc.Mobility fellow, funded by the Swiss National Science Foundation through the project 174987 H-Matrix Techniques and Uncertainty Quantification in Electromagnetism, the Excellence Initiative of the German Federal and State Governments and the Graduate School of Computational Engineering at TU Darmstadt. The work of F. Wolf is supported by DFG Grants SCHO1562/3-1 and KU1553/4-1 within the project Simulation of superconducting cavities with isogeometric boundary elements (IGA-BEM), the Excellence Initiative of the German Federal and State Governments and the Graduate School of Computational Engineering at TU Darmstadt.

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Appendix A

Appendix A

All the presented estimates are applicable to achieve three-dimensional estimates as well, going back to [3, 15]. We will briefly go over the construction and state the result corresponding to Theorem 2.

For \(p>0\) we define the spline complex on \([0,1]^3\) via

$$\begin{aligned} {\mathbb {S}}^0_{\pmb {p},\pmb {\varXi }}([0,1]^3)&:=S_{p_1,p_2,p_3}(\varXi _1,\varXi _2,\varXi _3),\nonumber \\ \pmb {{\mathbb {S}}}^1_{\pmb {p},\pmb {\varXi }}([0,1]^3)&:=S_{p_1-1,p_2,p_3}(\varXi _1',\varXi _2,\varXi _3)\nonumber \\&\quad \times \, S_{p_1,p_2-1,p_3}(\varXi _1,\varXi _2',\varXi _3) \nonumber \\&\quad \times \, S_{p_1,p_2,p_3-1}(\varXi _1,\varXi _2,\varXi _3'), \end{aligned}$$
(42)
$$\begin{aligned} \pmb {{\mathbb {S}}}^2_{\pmb {p},\pmb {\varXi }}([0,1]^3)&:=S_{p_1,p_2-1,p_3-1}(\varXi _1,\varXi _2',\varXi _3')\nonumber \\&\quad \times \, S_{p_1-1,p_2,p_3-1}(\varXi _1',\varXi _2,\varXi _3')\nonumber \\&\quad \times \, S_{p_1-1,p_2-1,p_3}(\varXi _1',\varXi _2',\varXi _3), \nonumber \\ {\mathbb {S}}^3_{\pmb {p},\pmb {\varXi }}([0,1]^3)&:=S_{p_1-1,p_2-1,p_3-1}(\varXi _1',\varXi _2',\varXi _3'). \end{aligned}$$
(43)

Let \(f_0,\)\(\pmb {f}_1,\)\(\pmb {f}_2,\)\(f_3\) be sufficiently smooth. We can use the transformations

$$\begin{aligned} \iota _0(\pmb {F})(f_0)&:=f_0\circ \pmb {F},\quad \iota _1(\pmb {F})(\pmb {f}_1):=(d\pmb {F})^\top (\pmb {f}_1\circ \pmb {F}),\nonumber \\ \iota _2(\pmb {F})(\pmb {f}_2)&:=\det (d\pmb {F}) (d\pmb {F})^{-1} (\pmb {f}_2\circ \pmb {F}),\quad \iota _3(\pmb {F})(f_3):=\det (d\pmb {F}) (f_3\circ \pmb {F}), \end{aligned}$$
(44)

to define the corresponding spaces in the single patch physical domain as in (3), cf. [32]. Now, the projections \(\tilde{\varPi }_{\pmb {p},\pmb {\varXi },\varOmega }^0\), \(\tilde{\pmb {\varPi }}_{\pmb {p},\pmb {\varXi },\varOmega }^{1}\), \(\tilde{\pmb {\varPi }}_{\pmb {p},\pmb {\varXi },\varOmega }^{2}\), and \(\tilde{\varPi }_{\pmb {p},\pmb {\varXi },\varOmega }^3\) w.r.t. the reference domain for \(\pmb {\varXi } = [\varXi _1,\varXi _2,\varXi _3]\) defined in complete analogy to (7), commute with the differential operators \(\pmb {{{\,\mathrm{grad}\,}}},\pmb {{{\,\mathrm{{curl}}\,}}}\) and \({\text {div}}\). By properties of the pullbacks, cf. [3, Sec. 5.1], this holds for the physical domain as well. The three-dimensional global B-spline projections are then defined as

$$\begin{aligned} \tilde{\varPi }^0_\varOmega&:=\bigoplus _{0\le j< N} \left( (\iota _{0}(\pmb {F}_j))^{-1}\circ \tilde{\varPi }_{\pmb {p},\pmb {\varXi },\varOmega }^0\circ \iota _{0}(\pmb {F}_j)\right) ,\nonumber \\ \tilde{\pmb {\varPi }}^1_\varOmega&:=\bigoplus _{0\le j< N} \left( (\iota _{1}(\pmb {F}_j))^{-1}\circ \tilde{\pmb {\varPi }}_{\pmb {p},\pmb {\varXi },\varOmega }^1\circ \iota _{1}(\pmb {F}_j)\right) ,\\ \tilde{\pmb {\varPi }}^2_\varOmega&:=\bigoplus _{0\le j< N} \left( (\iota _{2}(\pmb {F}_j))^{-1}\circ \tilde{\pmb {\varPi }}_{\pmb {p},\pmb {\varXi },\varOmega }^2\circ \iota _{2}(\pmb {F}_j)\right) ,\nonumber \\ \tilde{\varPi }^3_\varOmega&:=\bigoplus _{0\le j< N} \left( (\iota _{3}(\pmb {F}_j)^{-1}\circ \tilde{\varPi }_{\pmb {p},\pmb {\varXi },\varOmega }^3\circ \iota _{3}(\pmb {F}_j)\right) . \end{aligned}$$

In complete analogy to the Proof of Theorem 2, one can achieve the following result for the three-dimensional multipatch spline complex.

Corollary 5

Let the volumetric analogue of Assumptions 5 and 7 be satisfied. Assume the functions \(f_1,\)\(\pmb {f}_2,\)\(\pmb {f}_3,\)\(f_4\) to be sufficiently smooth, i.e., such that the norms and interpolation operators below are well defined. Then one finds, for integers s as below,

$$\begin{aligned} {\left\| f_1 - \tilde{\varPi }^0_\varOmega f_1 \right\| }_{H^r(\varOmega )}&\lesssim h^{s-r} {\left\| f_1 \right\| }_{{H}^s_{\mathrm {pw}}(\varOmega )},\quad \quad 3\le s\le p+1,\\ {\left\| \pmb {f}_2 - \tilde{\pmb {\varPi }}^1_\varOmega \pmb {f}_2 \right\| }_{\pmb {H}({{\,\mathrm{{curl}}\,}},\varOmega )}&\lesssim h^s {\left\| \pmb {f}_2 \right\| }_{{\pmb {H}}^s_{\mathrm {pw}}({{\,\mathrm{{curl}}\,}},\varOmega )},\quad 2< s\le p,\\ {\left\| \pmb {f}_3 - \tilde{\pmb {\varPi }}^2_\varOmega \pmb {f}_3 \right\| }_{\pmb {H}({\text {div}},\varOmega )}&\lesssim h^s {\left\| \pmb {f}_3 \right\| }_{{\pmb {H}}^s_{\mathrm {pw}}({\text {div}},\varOmega )},\quad 1 < s\le p,\\ {\left\| f_4 - \tilde{\varPi }^3_\varOmega f_4 \right\| }_{L^2(\varOmega )}&\lesssim h^s {\left\| f_4 \right\| }_{{H}^s_{\mathrm {pw}}(\varOmega )},\quad \quad \quad 0 \le s\le p, \end{aligned}$$

for \(r = 0,1\).

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Buffa, A., Dölz, J., Kurz, S. et al. Multipatch approximation of the de Rham sequence and its traces in isogeometric analysis. Numer. Math. 144, 201–236 (2020). https://doi.org/10.1007/s00211-019-01079-x

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Mathematics Subject Classification

  • 65D07
  • 65N12
  • 65N38