Discontinuous Galerkin Isogeometric Analysis for segmentations generating overlapping regions

TL;DR

This paper introduces a Discontinuous Galerkin Isogeometric Analysis method tailored for multipatch domains with small overlaps, providing theoretical error bounds and numerical validation for improved computational domain handling.

Contribution

The work develops a novel DG-IGA approach that effectively manages overlapping regions in multipatch geometries, with rigorous error analysis and practical verification.

Findings

The method accurately handles small overlaps in multipatch domains.

Theoretical error bounds are established and confirmed numerically.

Numerical examples demonstrate the method's effectiveness.

Abstract

In the Isogeometric Analysis (IGA) framework, the computational domain has very often a multipatch representation. The multipatch domain can be obtained by a volume segmentation of a boundary represented domain, e.g., provided by a Computer Aided Design (CAD) model. Typically, small gap and overlapping regions can appear at the patch interfaces of such multipatch representations. In the current work we consider multipatch representations having only small overlapping regions between the patches. We develop a Discontinuous Galerkin (DG)- IGA method which can be immediately applied to these representations. Our method appropriately connects the fluxes of the one face of the overlapping region with the flux of the opposite face. We provide a theoretical justification of our approach by splitting the whole error into two components: the first is related to the incorrect representation of the patches (consistency error) and the second to the approximation properties of the IGA space. We show bounds for both components of the error. We verify the theoretical error estimates in a series of numerical examples.

Tables

Table 1. Table 1: The values of the expected rates r 𝑟 \displaystyle r as they result from estimate ( 3.42 ).

	B-spline degree $p$
	Smooth solutions, $u\in H^{\ell \ge p+1}$
$d_o=h^{\lambda }$	$\lambda =1$	$\lambda =2$	$\lambda =2.5$	$\lambda =3$
$\beta :=$	0.5	1.5	2	2.5
$s:=$	$p$	$p$	$p$	$p$
$r:=$	$0.5$	$1.5$	$\min (p,\beta )$	$\min (p,\beta )$

Equations

$$-\mathrm{div}(\rho \nabla u)=f\text{in}\Omega \text{mbox}andu:=u_D=0\text{on}\partial \Omega ,$$

$$H^{\ell }(\Omega )=\{\varphi \in L^2(\Omega ):D^{\mathbf{\alpha }}\varphi \in L^2(\Omega ),\text{for all}|\alpha |\le \ell \},$$

$$\|\phi\|_{H^{\ell}(\Omega)}=\big{(}\sum_{0\leq|\boldsymbol{\alpha}|\leq\ell}\|D^{\boldsymbol{\alpha}}\phi\|_{L^{2}(\Omega)}^{2}\big{)}^{\frac{1}{2}}.$$

$$\displaystyle\displaystyle\hskip-5.69046pt{H^{1}_{0}(\Omega)=\{\phi\in H^{1}(\Omega):\phi=0\,\text{on}\,\partial\Omega\}},\quad H^{1}_{\Gamma}(\Omega)=\{\phi\in H^{1}(\Omega):\phi=0\,\text{on}\,{\color[rgb]{0,0,0}\Gamma\subset\partial\Omega,\,|\Gamma|>0.}\}.$$

$$\left|{\int }_{\Omega }{\varphi }_1{\varphi }_{2}dx\right|\le \Vert {\varphi }_1{\Vert }_{L^2(\Omega )}\Vert {\varphi }_2{\Vert }_{L^2(\Omega )}\text{mbox}and\left|{\int }_{\Omega }{\varphi }_1{\varphi }_{2}dx\right|\le \frac{ϵ}{2}\Vert {\varphi }_1{\Vert }_{L^2(\Omega )}^2+\frac{1}{2ϵ}\Vert {\varphi }_2{\Vert }_{L^2(\Omega )}^2,$$

$$\begin{array}{rl}\Vert \varphi {\Vert }_{L^2(\partial \Omega )}^2\le & C_{tr}\Vert \varphi {\Vert }_{L^2(\Omega )}\Vert \varphi {\Vert }_{H^1(\Omega )},\\ \Vert \varphi {\Vert }_{L^2(\Omega )}\le & {\text{meas}}_{{\mathbb{R}}^d}(\Omega )\Vert \nabla \varphi {\Vert }_{L^2(\Omega )},\text{for}\varphi \in H_{\Gamma }^1(\Omega ).\end{array}$$

$$a(u,\varphi )=l_f(\varphi ),\forall \varphi \in H_0^1(\Omega ),$$

$$a(u,\varphi )={\int }_{\Omega }\rho \nabla u\cdot \nabla \varphi dx\text{mbox}and{l}_f(\varphi )={\int }_{\Omega }f\varphi dx,$$

$$\Xi =\{\underset{=z_1}{\underbrace{0={\xi }_1,\dots ,{\xi }_{m_1}}},\underset{=z_2}{\underbrace{{\xi }_{m_1+1}=\dots ={\xi }_{m_1+m_2}}},\dots ,\underset{=z_M}{\underbrace{{\xi }_{n_1+p+1-m_M},\dots ,{\xi }_{n_1+p+1}=1}}\}.$$

$$B_{i,p}=$$

$$\text{with}{B}_{i,0}(x)=$$

$${\hat{\mathbb{B}}}_{\mathbf{\Xi },p}={\otimes }_{k=1}^d{\hat{\mathbb{B}}}_{{\Xi }^k,p}=\text{span}\{{\hat{B}}_j(\hat{x}){\}}_{j=1}^{n=n_1\cdot \dots \cdot n_k\cdot \dots \cdot n_d},$$

$${\hat{B}}_j(\hat{x})=$$

$$\mathbf{\Phi }:\hat{\Omega }\to \Omega ,x=\mathbf{\Phi }(\hat{x})=\sum _{j=1}^n{\mathbf{C}}_j{\hat{B}}_j(\hat{x})\in \Omega ,$$

$$\displaystyle\displaystyle\mathbb{B}_{\mathbf{\Xi},p}:={\color[rgb]{0,0,0}\text{span}}\{B_{{j}}|_{\Omega}:B_{j}({x})=\hat{B}_{j}\circ\mathbf{\Phi}^{-1}({x}),{\ }\text{for}{\ }\hat{B}_{j}\in\hat{\mathbb{B}}_{\mathbf{\Xi},p}\}.$$

$$\overline{\Omega }={\cup }_{i=1}^N{\overline{\Omega }}_i,\text{with}{\Omega }_i\cap {\Omega }_j=\varnothing ,\text{for}i\ne j,$$

$$V_{\mathbb{B}}:=\{{\varphi }_h\in L^2(\Omega ):{\varphi }_h{|}_{{\Omega }_i}\in {\mathbb{B}}_{{\mathbf{\Xi }}_i,p}\}.$$

$$\overline{\Omega }=\overline{{\Omega }_1}\cup \overline{{\Omega }_2},$$

$$[[u]]:=u_1-u_2=0\text{ on }{F}_{12},\text{and}[[\rho \nabla u]]\cdot n_{F_{12}}:=({\rho }_1\nabla u_1-{\rho }_2\nabla u_2)\cdot n_{F_{12}}=0\text{ on }{F}_{i12},$$

$$H^{\ell }({\mathcal{T}}_H(\Omega ))=\{u\in L^2(\Omega ):u_i=u{|}_{{\Omega }_i}\in H^{\ell }({\Omega }_i),\text{for}i=1,2\}.$$

$$\overline{\Omega }=\overline{{\Omega }_1^{\ast }}\cup \overline{{\Omega }_2^{\ast }},$$

$$0\le x\le x_{M_o},0\le y\le y_{M_o},z={\zeta }_0(x,y),$$

$$x_{o1}\in F_{o12}\to {\mathbf{\Phi }}_{o12}(x_{o1}):=x_{o2}\in F_{o21},\text{with }{\mathbf{\Phi }}_{o12}(x_{o1})=x_{o1}+{\zeta }_o(x_{o1})n_{F_{o12}},$$

$$d_o=$$

$$d_o\le$$

$${\mathbf{\Phi }}_{o21}(x_{o2})=x_{o1},\text{with}{\mathbf{\Phi }}_{o12}(x_{o1})=x_{o2},$$

$$\begin{array}{rl}{H}^{\ell }({\mathcal{T}}_H^{\ast }(\Omega )):=& \{u^{\ast }=(u_1^{\ast },u_2^{\ast }):u_i^{\ast }\in H^{\ell }({\Omega }_i^{\ast }),u_i^{\ast }{|}_{\partial {\Omega }_i^{\ast }\cap \partial \Omega }=0,\text{for}i=1,2\},\\ H_0^{\ell }({\mathcal{T}}_H^{\ast }(\Omega )):=& \{u^{\ast }=(u_1^{\ast },u_2^{\ast }):u_i^{\ast }\in H_0^{\ell }({\Omega }_i^{\ast }),\text{for}i=1,2\},\\ V_{\mathbb{B}}^{\ast }:=& \{{\varphi }_h^{\ast }=({\varphi }_{1,h},{\varphi }_{2,h}):{\varphi }_{i,h}\in {\mathbb{B}}_{{\mathbf{\Xi }}_i,p}^{\ast },\text{for}i=1,2\}.\end{array}$$

$$\displaystyle\displaystyle\hskip 0.0pt{\|v\|^{2}_{DG}=\sum_{i=1}^{2}\Big{(}\rho_{i}\|\nabla v_{i}\|^{2}_{L^{2}(\Omega^{*}_{i})}+\frac{\rho_{i}}{h}\|v_{i}\|^{2}_{L^{2}(\partial\Omega^{*}_{i}\cap\partial\Omega)}+\sum_{F_{oij}\subset\partial\Omega_{i}^{*}}\frac{\{\rho\}}{h}\|v_{i}\|^{2}_{L^{2}(F_{oij})}\Big{)},{\ }\text{for}{\ }1\leq i\neq j\leq 2,}$$

$$(P1)\{\begin{array}{ll}-\mathrm{div}({\rho }_1\nabla u_1^{\ast })& =f,\text{in}{\Omega }_1^{\ast },\\ u_1^{\ast }=u_D& \text{on}\partial {\Omega }_1^{\ast }\cap \partial \Omega ,\\ u_1^{\ast }=u& \text{on}{F}_{o12},\end{array}$$

$$(P2)\{\begin{array}{ll}-\mathrm{div}({\rho }_2\nabla u_2^{\ast })& =f,\text{in}{\Omega }_2^{\ast },\\ u_2^{\ast }=u_D& \text{on}\partial {\Omega }_2^{\ast }\cap \partial \Omega ,\\ u_2^{\ast }=u& \text{on}{F}_{o21},\end{array}$$

$$u_1^{\ast }\in H_{\partial {\Omega }_1^{\ast }\cap \partial \Omega }^1({\Omega }_1^{\ast })$$

Full text

1 Introduction

Isogeometric Analysis (IGA) has been introduced in [18] as a new methodology for solving numerically Partial Differential Equations (PDE). The key idea of the IGA concept is to use the superior finite dimensional spaces, which are used in Computer Aided Design (CAD), e.g., B-splines, NURBS, for both the exact representation of the computational domain $\displaystyle\Omega$ and discretizing the PDE problem. Since this work, many applications of the IGA methodology to several fields have been discussed in several papers, see, e.g., the monograph [6] and the references within, as well as the survey paper [7]. From a computational point of view, we can say that the numerical algorithm for constructing the B-spline (or NURBS) basis functions is quite simple. This helps to produce high order approximate solutions. From the theoretical point of view, the fundamental approximation properties of the B- spline spaces on a reference domain are discussed in [30]. The approximation properties of the mapped B-spline (or NURBS) spaces are discussed in several papers, see e.g., [2], [32], [7], [22].

Let us consider a complex domain $\displaystyle\Omega$ where its boundary is prescribed by CAD models. The CAD models can not be directly used in IGA in order to discretise the PDE problems. We need to create volumetric patch parametrizations from the CAD models. The boundary represented domain is first segmented into a collection of suitable blocks and consequently a parametrization procedure is applied to each block. This produces the volumetric multipatch representation $\displaystyle\cup_{i=1}^{N}\overline{\Omega_{i}}$ of $\displaystyle\overline{\Omega}$ suitable for IGA. Several segmentation algorithms and associated parametrization procedures have been discussed in the literature, see, e.g., [17], [27],[19], [34, 35]. Furthermore, we refer to [33], [10] and [5] for different approaches for constructing IGA planar parametrizations without utilizing segmentation algorithms. We mention the segmentation approach presented in [24], and [25], from which, we have been motivated to present the current work. The main idea is to split the given boundary represented domain, using a spline curve (or face in 3d case) with the following properties: (i) must have the end points on the boundary and the tangents to be specified, (ii) the curve is reasonably regular and does not intersect the boundary of the domain, (iii) the curve cuts the domain into new subdomains with good shapes. Consequently, tensor-product B-spline spaces are fitted in the collection of the subdomains for defining the tensor product B-spline surfaces or volumes [10]. Note that the previous consideration is also concerns CAD models that are connected along a non-matching interface. It is important to obtain a curve that splits $\displaystyle\Omega$ into new simple domains with good shapes being suitable for IGA. During the computation of the multipatch representation, errors can occur when defining the corresponding control points, see [27], [24] and [10]. A consequence of this is a non-conforming parametrizations of the patches in the sense that the images of the patch interfaces under the parametrizations are not identical. This in turn leads to the existence of gap and/or overlapping regions between the adjoining patches, see a schematic illustration in Fig. 1(b).

This paper considers the case where there are only overlapping regions between the patches. If we apply an IGA methodology to this multipatch representation, a direct consequence is that the whole discretization error will include two (main) parts: the first naturally comes from the approximation properties of the B-spline spaces (for the purposes of this work we use B-spline spaces) and, the second comes from the geometric error. The later is due to the incorrect parametrization of the patch interfaces. Furthermore, the geometric error can be characterized as a consistency error, which consists of two error components. The first error component is related to the approximation of the jumps of the flux of the solution on the non-matching interfaces. The second component is related to the existence of more than one numerical solution in the overlapping regions.

The contribution of this paper is to develop a DG-IGA method which can be applied on volumetric patch representations with non-matching interface parametrizations. We present our methodology for discretizing the following elliptic Dirichlet boundary value problem

[TABLE]

where the diffusion coefficient $\displaystyle\rho(x)$ can be discontinuous across a smooth internal interface. We derive bounds for the two main parts of the whole error. In our analysis, we derive separate bounds for the two components of the geometric error. To the best of our knowledge, we believe this is an new area of analysis to be investigated. Our current work is the first step in the analysis, where we are developing our methodology for the numerical solution of the simple stationary diffusion problem (1.1). Our intention for future works is to extend the current methodology to more complicated time dependent problems, where the interface can move with time, cf. [4].

Due to the non-matching interior patch interfaces, a direct application of the classical DG numerical fluxes proposed in literature, see e.g. [22], [26], is not possible, as these fluxes are only applicable for matching interface parametrizations. In our recent papers, [14] and [16], we developed DG-IGA schemes for multipatch unions that include only gap regions. In particular, we considered the PDE model given in (1.1) and we denoted by $\displaystyle d_{g}$ the maximum distance between the diametrically opposite points located on the gap boundary. We applied Taylor expansions using the diametrically opposite points of the gap, in order to give estimates for the jumps of the normal fluxes with respect to $\displaystyle d_{g}$. Finally, we used the same Taylor expansions in the DG-IGA scheme for constructing suitable DG numerical fluxes across the gap boundary that help on the weakly coupling of the local patch-wise discrete problems. We developed a discretization error analysis and showed a priori estimates in the DG-norm, expressed in terms of the mesh size and the gap width, i.e., $\displaystyle\mathcal{O}(h^{r})+\mathcal{O}(d_{g})$, where $\displaystyle r$ depends on the B-spline degree $\displaystyle p$ and the regularity of the solution. In [14] and [16], we have shown that, if $\displaystyle d_{g}=\mathcal{O}(h^{p+\frac{1}{2}})$, the proposed DG-IGA scheme has optimal approximation properties.

In this paper, we extend the previous work to multipatch unions with overlapping regions. In the analysis presented in [14] and [16], the whole geometric error does not include the component coming from the co-existence of different IGA solutions in the overlapping regions. Here, the new approach is to introduce local (patch-wise) auxiliary variational problems, which are compatible with the overlapping nature of the multipatch representation of $\displaystyle\Omega$. We denote the solutions of the new variational problems by $\displaystyle u^{*}$. These problems are not consistent, in the sense that the original solution $\displaystyle u$ of (1.1) does not satisfy them. Following the IGA concept, the B-spline spaces used for the parametrization of the patches are also used for discretizing the local auxiliary problems. We denote by $\displaystyle u_{h}^{*}$ the produced IGA solutions. Under some regularity assumptions on $\displaystyle u^{*}$, we can expect (see Section 3) that the IGA solution $\displaystyle u_{h}^{*}$ has optimal approximation properties associated with $\displaystyle u^{*}$. However, we can not directly infer that $\displaystyle u_{h}^{*}$ can approximate in an optimal way the solution $\displaystyle u$ of the original problem. In our analysis, we provide an estimate for the consistency error $\displaystyle u-u^{*}$ and consequently using the triangle inequality $\displaystyle\|u-u_{h}^{*}\|_{DG}\leq\|u^{*}-u_{h}^{*}\|_{DG}+\|u-u^{*}\|_{DG}$, we can derive an estimate for the error between the exact solution $\displaystyle u$ and the IGA solution $\displaystyle u_{h}^{*}$. The mesh-dependent norm $\displaystyle\|\cdot\|_{DG}$ is defined in Section 2. We give error estimates for both terms $\displaystyle\|u^{*}-u_{h}^{*}\|_{DG}$ and $\displaystyle\|u-u^{*}\|_{DG}$ expressed in terms of the mesh size $\displaystyle h$ and the quantity $\displaystyle d_{o}$, which is introduced in our analysis in order to quantify the width of the overlapping regions. In particular, we show that under appropriate assumptions on the data and for the case where $\displaystyle d_{o}$ is of order $\displaystyle\textstyle{h^{\lambda},\,\lambda\geq p+\frac{1}{2}}$, the proposed DG-IGA scheme has optimal convergence properties. This convergence result is similar to the result in [14] and [16].

In a future work, we apply the same approach to solve problems on multipatch partitions, which can include gap and overlapping regions. We present numerical solutions in multipatch unions with more complicated gaps and overlapping regions. We also provide details related to the implementation of the proposed DG-IGA scheme. In the same work, we also discuss issues related to the construction of domain decomposition methods on these multipatch representations and provide several numerical tests for evaluating their performance. The first results in this direction can be found in [15].

We note that IGA multipatch representations with non-matching interfaces meshes, overlapping regions and trimmed patches have been considered in many publications. For the communication of the discrete patch-wise problems, several Nitsche’s type coupling methods involving normal flux terms have been applied across the interfaces, see e.g., [29],[26],[1],[3] and references therein. We mention also that in [36], DG-IGA methods have been presented to discretize Laplace problems on multipatch unions with large overlapping regions. The proposed strategy follows the additive Schwartz methodology. To the knowledge of the authors, there are no works that analytically discuss estimates for the error, which is caused by the incorrect representation of the shape of the patches. The purpose of this work is to present such an error analysis.

The structure of the paper is as follows: Section 2 presents the PDE model, briefly reviews the B-spline spaces and describes the case of having non-matching parametrized interfaces with overlapping regions. Section 3, presents in detail the perturbation problems, the bounds for the consistency error, the proposed DG-IGA scheme and the error analysis. Section 4, includes several numerical examples that confirm the theoretical estimates. The paper closes with the Conclusions.

2 The model problem

2.1 Preliminaries

Let $\displaystyle\Omega$ be a bounded Lipschitz domain in $\displaystyle\mathbb{R}^{d},\,d=2,3$, and let $\displaystyle\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{d})$ be a multi-index of non-negative integers $\displaystyle\alpha_{1},\ldots,\alpha_{d}$ with degree $\displaystyle\textstyle{|\boldsymbol{\alpha}|=\sum_{j=1}^{d}\alpha_{j}}$. For any $\displaystyle\boldsymbol{\alpha}$, we define the differential operator $\displaystyle D^{\boldsymbol{\alpha}}=D_{1}^{\alpha_{1}}\ldots D_{d}^{\alpha_{d}}$, with $\displaystyle D_{j}=\partial/\partial x_{j}$, $\displaystyle j=1,\ldots,d$, and $\displaystyle D^{(0,\ldots,0)}\phi=\phi$. For a non-negative integer $\displaystyle m$, let $\displaystyle C^{m}(\Omega)$ denote the space of all functions $\displaystyle\phi:\Omega\rightarrow\mathbb{R}$, whose partial derivatives $\displaystyle D^{\boldsymbol{\alpha}}\phi$ of all orders $\displaystyle|\boldsymbol{\alpha}|\leq m$ are continuous in $\displaystyle\Omega$. Let $\displaystyle\ell$ be a non-negative integer. As usual, $\displaystyle L^{2}(\Omega)$ denotes the Sobolev space for which $\displaystyle\textstyle{\int_{\Omega}|\phi(x)|^{2}\,dx<\infty}$, endowed with the norm \displaystyle\textstyle{\|\phi\|_{L^{2}(\Omega)}=\big{(}\int_{\Omega}|\phi(x)|^{2}\,dx\big{)}^{\frac{1}{2}}}, and $\displaystyle L^{\infty}(\Omega)$ denotes the functions that are essentially bounded. Also

[TABLE]

denote the standard Sobolev spaces endowed with the following norms

[TABLE]

We identify $\displaystyle L^{2}$ and $\displaystyle H^{0}$ and also define the subspace $\displaystyle H^{1}_{0}(\Omega)$ and $\displaystyle H^{1}_{\Gamma}(\Omega)$ of $\displaystyle H^{1}(\Omega)$

[TABLE]

We recall Hölder’s and Young’s inequalities

[TABLE]

that hold for all $\displaystyle\phi_{1}\in L^{2}(\Omega)$ and $\displaystyle\phi_{2}\in L^{2}(\Omega)$ and for any fixed $\displaystyle\epsilon\in(0,\infty)$. In addition, we recall trace and Poincare’s inequalities, [9],

[TABLE]

2.2 The elliptic diffusion problem

The weak formulation of the boundary value problem (1.1) reads as follows: for given source function $\displaystyle f\in L^{2}(\Omega)$ find a function $\displaystyle u\in H_{0}^{1}(\Omega)$ such that the variational identity

[TABLE]

is satisfied, where the bilinear form $\displaystyle a(\cdot,\cdot)$ and the linear form $\displaystyle l_{f}(\cdot)$ are defined by

[TABLE]

respectively.

The given diffusion coefficient $\displaystyle\rho\in L^{\infty}(\Omega)$ is assumed to be uniformly positive and piece-wise (patch-wise, see below) constant. These assumptions ensure existence and uniqueness of the solution due to Lax-Milgram’s lemma.

For simplicity, we only consider pure Dirichlet boundary conditions on $\displaystyle\partial\Omega$. However, the analysis presented in our paper can easily be generalized to other constellations of boundary conditions which ensure existence and uniqueness such as Robin or mixed boundary conditions.

In what follows, positive constants $\displaystyle c$ and $\displaystyle C$ appearing in inequalities are generic constants which do not depend on the mesh-size $\displaystyle h$. In many cases, we will indicate on what may the constants depend on. Frequently, we will write $\displaystyle a\sim b$ meaning that $\displaystyle c\,a\leq b\leq C\,a$.

2.3 B-spline spaces

In this section, we briefly present the B-spline spaces and the form of the B-spline parametrizations for the physical subdomains. For a better presentation of the B-spline spaces, we start our discussion for the one-dimensional case. Then we proceed to higher dimensions. We refer to [6], [8] and [30] for a more detailed presentation.

Consider, $\displaystyle\mathcal{Z}=\{0=z_{1}<z_{2}<\ldots<z_{M}=1\}$ to be a partition of $\displaystyle\bar{I}=[0,1]$ with $\displaystyle\bar{I}_{j}=[z_{j},z_{j+1}],\,j=1,\ldots,M-1$ to be the intervals of the partition. Let the integers $\displaystyle p$ and $\displaystyle n_{1}$ denote the $\displaystyle p$ spline degree and the number of the B-spline basis functions.

Based on $\displaystyle\mathcal{Z}$, we introduce the open knot vector $\displaystyle\Xi=\{0=\xi_{1},\xi_{2},\ldots,\xi_{n_{1}+p+1}=1\}$, and the associated vector $\displaystyle\mathcal{M}=\{m_{1},\ldots,m_{M}\}$ of knot multiplicities with $\displaystyle m_{1}=m_{M}=p+1$, i.e.,

[TABLE]

The B-spline basis functions are defined by the Cox-de Boor formula, see, e.g., [6] and [8],

[TABLE]

We assume that $\displaystyle m_{j}\leq p$ for all internal knots, which in turn gives that, at $\displaystyle z_{j}$ the B-spline basis functions have $\displaystyle\mathcal{\kappa}_{j}=p-m_{j}$ continuous derivatives.

Let us now consider the unit cube $\displaystyle\widehat{\Omega}=(0,1)^{d}\subset\mathbb{R}^{d}$, which we will refer to as the parametric domain. Let the integers $\displaystyle p$ and $\displaystyle n_{k}$ denote the given B-spline degree and the number of basis functions of the B-spline space that will be constructed in $\displaystyle x_{k}$-direction with $\displaystyle k=1,\ldots,d$. We introduce the $\displaystyle d-$dimensional vector of knots $\displaystyle\mathbf{\Xi}=(\Xi^{1},\ldots,\Xi^{k},\ldots,\Xi^{d}),$ with the particular components given by $\displaystyle\Xi^{k}=\{0=\xi^{k}_{1},\xi^{k}_{2},\ldots,\xi^{k}_{n_{k}+p+1}=1\}$,  $\displaystyle k=1,\ldots,d$, .

Given the knot vector $\displaystyle\Xi^{k}$ in every direction $\displaystyle k=1,\ldots,d$, we construct the associated univariate B-spline basis functions, $\displaystyle\hat{\mathbb{B}}_{\Xi^{k},p}=\{\hat{B}_{1,k}(\hat{x}_{k}),\ldots,\hat{B}_{n_{k},k}(\hat{x}_{k})\}$, see, e.g., [8] for more details. Accordingly, the B-spline basis functions of $\displaystyle\hat{\mathbb{B}}_{\mathbf{\Xi},k}$ are defined by the tensor-product of the univariate B-spline basis functions, that is

[TABLE]

where each $\displaystyle\hat{B}_{j}(\hat{x})$ has the form

[TABLE]

In the IGA framework, the computational domain $\displaystyle\Omega$ is described as the image of $\displaystyle\widehat{\Omega}$ under a B-spline, NURBS, etc., parametrization mapping of the form

[TABLE]

where $\displaystyle\mathbf{C}_{j},\,j=1,\ldots,n$ are the control points and $\displaystyle\hat{x}=\mathbf{\Phi}^{-1}(x)$, see Fig. 1(a). Following the IGA methodology, [18], [6], the B-spline spaces for discretizing the PDE problem are defined by using the mapping given in (2.9), i. e., we define the B-spline space in $\displaystyle\Omega$ by

[TABLE]

2.3.1 Multipatch representations and B-spline spaces

Our contribution here aims at developing a DG-IGA method appropriate for discretizing PDE models on non-conforming multipatch partitions of the domain $\displaystyle\Omega$. Let us suppose that the domain $\displaystyle\Omega$ is described as a union of N-subdomains

[TABLE]

with interior interfaces $\displaystyle F_{ij}=\partial\Omega_{i}\cap\partial\Omega_{j}$, for $\displaystyle 1\leq i\neq j\leq N$. We further suppose that every subdomain $\displaystyle\Omega_{i}$ has its own parametrization $\displaystyle\mathbf{\Phi}_{i}$, which is defined by the corresponding B-spline space $\displaystyle\hat{\mathbb{B}}_{\mathbf{\Xi}_{i},p}$ and the corresponding control points $\displaystyle\mathbf{C}_{j}^{(i)}$, see (2.9). Here $\displaystyle\mathbf{\Xi}_{i}$ denotes the knot-vector related to $\displaystyle\Omega_{i}$. An illustration for $\displaystyle N=2$ is given in Fig. 1(a). The subdomains $\displaystyle\Omega_{i}$ are referred to as patches. In an analogous way as in (2.10), we define the physical patch-wise B-spline spaces $\displaystyle\mathbb{B}_{\mathbf{\Xi}_{i},p}$ for $\displaystyle i=1,\ldots,N$. We define the global discontinuous B-spline space $\displaystyle V_{\mathbb{B}}$ with components on every $\displaystyle\mathbb{B}_{\mathbf{\Xi}_{i},p}$

[TABLE]

Assumption 1

Assume that every $\displaystyle\mathbf{\Phi}_{i},\,i=1,...,N$ is sufficiently smooth and there exist constants $\displaystyle 0<c<C$ such that $\displaystyle c\leq|\det J_{\mathbf{\Phi}_{i}}|\leq C$, where $\displaystyle J_{\mathbf{\Phi}_{i}}$ is the Jacobian matrix of $\displaystyle\mathbf{\Phi}_{i}$.

The components of $\displaystyle\mathbf{\Xi}_{i}$ form a mesh $\displaystyle T^{(i)}_{h_{i},\widehat{\Omega}}=\{\hat{E}_{m}\}_{m=1}^{M_{i}}$ in $\displaystyle\widehat{\Omega}$, where $\displaystyle\hat{E}_{m}$ are the micro-elements and $\displaystyle h_{i}$ is the mesh size, which is defined as follows. Given an element $\displaystyle\hat{E}_{m}\in T^{(i)}_{h_{i},\widehat{\Omega}}$, we set $\displaystyle h_{\hat{E}_{m}}=\text{diameter}(\hat{E}_{m})$ and the mesh size $\displaystyle h_{i}$ is defined to be $\displaystyle h_{i}=\max\{h_{\hat{E}_{m}}\}$. We set $\displaystyle\textstyle{h=\max_{i=1,\ldots,N}\{h_{i}\}}$. For every $\displaystyle\Omega_{i}$, we construct a mesh $\displaystyle T^{(i)}_{h_{i},\Omega_{i}}=\{E_{m}\}_{m=1}^{M_{i}}$, whose vertices are the images of the vertices of the corresponding parametric mesh $\displaystyle T^{(i)}_{h_{i},\widehat{\Omega}}$ under $\displaystyle\mathbf{\Phi}_{i}$.

Assumption 2

The meshes $\displaystyle T^{(i)}_{h_{i},\widehat{\Omega}}$ are quasi-uniform, i.e., there exist a constant $\displaystyle\theta\geq 1$ such that $\displaystyle\theta^{-1}\leq{h_{\hat{E}_{m}}}/{h_{\hat{E}_{m+1}}}\leq\theta$. Also, we assume that $\displaystyle h_{i}\sim h_{j}$ for $\displaystyle 1\leq i\neq j\leq N$.

2.4 Multipatch description of the computational domain

In many practical applications, the parametrization of a boundary represented domain $\displaystyle\Omega$ by a single B-spline (NURBS) patch may not be posible. In order to discretize a PDE problem following the IGA framework in this situation, we represent the domain $\displaystyle\Omega$ as a multipatch. Following the methodology presented in [24, 19], the initial domain $\displaystyle\Omega$ is firstly segmented into a collection of simple subdomains, e.g., topological hexahedra. Consequently, a suitable parametrization mapping is constructed for each subdomain for obtaining the multipatch representation of $\displaystyle\Omega$. The final parametrization mappings of the adjoining patches must provide identical images for the common interfaces. In particular, for a DG-IGA discretization of the model (1.1), it would be preferable to produce a multipatch partition of $\displaystyle\Omega$ compatible with the variations of the coefficient $\displaystyle\rho$, i.e., the patches to be coincided with the parts of $\displaystyle\Omega$ where the coefficient $\displaystyle\rho$ is constant.

For example, let us consider Fig. 1(a). In this case the domain $\displaystyle\Omega$ is described as a union of two non overlapping patches, see (2.11), i.e.,

[TABLE]

where the interface $\displaystyle F_{12}$ coincides with the physical interface. We use the notation $\displaystyle\mathcal{T}_{H}(\Omega):=\{\Omega_{1},\Omega_{2}\}$ for the union (2.13). For each $\displaystyle\Omega_{i},\,i=1,2$, there exists a matching parametrization mapping such that $\displaystyle\mathbf{\Phi}_{i}:\widehat{\Omega}\rightarrow\Omega_{i}$ with $\displaystyle\Omega_{i}=\mathbf{\Phi}_{i}(\widehat{\Omega})$. The control points, which are related to the patch interface $\displaystyle F_{12}$, are appropriately matched in order for the parametrizations $\displaystyle\mathbf{\Phi}_{1}$ and $\displaystyle\mathbf{\Phi}_{2}$ of the neighboring patches to give the same image for the parametrized interface $\displaystyle F_{12}$. Based on $\displaystyle\mathcal{T}_{H}(\Omega)$, we can independently discretize the problem on the different patches $\displaystyle\Omega_{i},\,i=1,2$, using interface conditions across $\displaystyle F_{12}$ for coupling the local problems. Typically, the interface conditions across $\displaystyle F_{12}$ concern continuity requirements of the solution $\displaystyle u$ of (1.1), i.e.,

[TABLE]

where $\displaystyle n_{F_{12}}$ is the unit normal vector on $\displaystyle F_{12}$ with direction towards $\displaystyle\Omega_{2}$, and $\displaystyle\rho_{i},\,u_{i},\,i=1,2$