Abstract
In applied computations the need arises to define, for example, a discrete field with assigned curl or to represent a div-free field in a given discrete space. In the low degree case this need is often fulfilled by employing tree and co-tree techniques. The definition of tree and co-tree is thus revisited here in the frame of high order Whitney element reconstructions. We consider the case of fields that are reconstructed in a contractible polyhedral domain \(\varOmega \in \mathbb R^3\), with connected boundary ∂Ω, starting from their weights over suitable “small simplices” in a simplicial mesh \({\mathcal M}\) of the domain \(\bar \varOmega \).
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1 Introduction
We aim at determining in a constructive way, for the high order case, the finite element solutions of grad ϕ = E, curl A = B, div D = ρ, namely, of the equations linking the electric field E, the magnetic induction B, and the electric charge density ρ, to their potentials ϕ, A and D, respectively. Stating the necessary and sufficient conditions for assuring that a function defined in a bounded set \(\varOmega \subset {\mathbb R}^3 \) is the gradient of a scalar potential, the curl of a vector potential or the divergence of a vector field is one of the most classical problem of vector analysis (see for example [4, 8, 10]). We aim at providing an explicit and efficient procedure to construct a finite element solution. For example, div-free fields, W, are implicitly characterized in terms of a vector w of degrees of freedom of W by the algebraic constraint D w = 0, with D the matrix of the div operator between finite elements spaces. The same fields, in the case of a domain with connected boundary, are explicitly defined by w = R a, with no constraint on a, where R is the matrix of the curl operator between finite elements spaces and a collects the degrees of freedom of the vector potentials A. Similarly, one can wish to compute a vector potential a such that R a = b, for a given field b verifying D b = 0. As explained in [6], these bases can be constructed by the help of “trees” and “co-trees”, which are at the core of this contribution. The case r = 0 is largely treated in the literature for different types of topological domains (see for example [2]). In these pages, we develop the tree and co-tree approaches for r > 0 when fields in the high order Whitney spaces are represented on the basis of their weights on small simplices [7, 9, 12]. With this choice of degrees of freedom, the tree and co-tree concepts extend from r = 0 to r > 0 straightforwardly.
2 Basic Concepts
Let \(\varOmega \subset {\mathbb R}^3\) be a bounded polyhedral domain with Lipschitz boundary ∂Ω and \({\mathcal M}\) a simplicial mesh of \(\bar \varOmega \). We denote by |A| the cardinality of the set A. For 0 ≤ k ≤ 3, let Δ k(T) (resp. \(\varDelta _k({\mathcal M})\)) be the set of k-simplices of a mesh tetrahedron T (resp. of the mesh \({\mathcal M}\)). Note that \(\varDelta _k({\mathcal M}) = \cup _{T \in {\mathcal M}} \varDelta _k(T)\). If \(\varDelta _0({\mathcal M}) = \{ {\mathbf {v}}_i\}_{i}\), with i = 1, …, N v, being \(N_v = |\varDelta _0({\mathcal M})|\), then each k-simplex \(S \in \varDelta _k({\mathcal M})\) has associated an increasing map m S : {0, …, k} → {1, …, N v}. This map induces an (inner) orientation on S (i.e., a way to run along S if k = 1, through S if k = 2, in S if k = 3).
If we assign to each \(S \in \varDelta _k({\mathcal M}) \) a real number c S we can define the k-chain \(c = \sum _{S \in \varDelta _0({\mathcal M}) } {c_S} \, S\), i.e. a formal weighted sum of k-simplices S in \({\mathcal M}\). One can add k-chains, namely \((c + \tilde c) = \sum _S ({c_S + \tilde c_S})\,S\), and multiply a k-chain by a scalar p, namely p c =∑S(p c S) S. The set of all k-chains in \({\mathcal M}\), here denoted \({\mathcal C}_k({\mathcal M})\), is a vector space, in one-to-one correspondence with the set of real vectors \({c} = ({ c}_S)_{S \in \varDelta _k({\mathcal M})}\). Each k-simplex \(S \in \varDelta _k({\mathcal M}) \), can be associated with the elementary k-chain c with entries c S = 1 and \({ c}_{\tilde S} = 0\) for \(\tilde S \neq S\). In the following we will use the same symbol S to denote the oriented k-simplex and the associated elementary k-chain.
The boundary operator ∂ takes a k-simplex S and returns the sum of all its (k − 1)-faces f with coefficient 1 or − 1 depending of whether the orientation of the (k − 1)-face f matches or not with the orientation induced by that of the simplex S on f. Since the boundary operator is a linear mapping from \({\mathcal C}_k({\mathcal M})\) to \({\mathcal C}_{k-1}({\mathcal M})\), it can be represented by a matrix ∂ of dimension \(|\varDelta _{k-1}({\mathcal M}) | \times |\varDelta _k({\mathcal M}) |\), which is rather sparse, gathering the coefficients 0, − 1, or + 1. Note that in three dimensions, there are three nontrivial boundary operators acting, respectively, on edges, triangles and tetrahedra: ∂ 1 represented by the matrix G ⊤, ∂ 2 represented by R ⊤, and ∂ 3 represented by D ⊤. To fully specify ∂, we need to specify the boundary of each simplex S. By definition, we have
for any \(e \in \varDelta _1({\mathcal M})\), any \(f \in \varDelta _2({\mathcal M})\) and any \(T \in \varDelta _3({\mathcal M})\). For e = [v 0, v 1], f = [v 0, v 1, v 2] and T = [v 0, v 1, v 2, v 3], we have, respectively,
The subscript is removed when there is no ambiguity, since the operator needed for a particular operation is indicated from the type of the operand (e.g., ∂ 3 when ∂ applies to tetrahedra). The notion of boundary can be extended to k-chains by linearity, \(\partial c = \partial (\sum _{S \in \varDelta _k({\mathcal M}) } {\mathbf {c}}_S \, S) = \sum _{S \in \varDelta _k({\mathcal M}) } {\mathbf {c}}_S\, \partial S \).
We say that a k-chain c is closed if ∂ k c = 0. Non-trivial closed k-chains are called k-cycles and constitute the subspace \(Z_k({\mathcal M}) = \mathrm {ker}(\partial _k; \mathcal C_k({\mathcal M}))\). A k-chain c is a boundary if it exists a (k + 1)-chain γ such that c = ∂ k+1 γ. The k-boundaries constitute the subspace \(B_{k}({\mathcal M}) = \partial _{k+1} \mathcal C_{k+1}({\mathcal M})\). From the property ∂∂ = 0, we know that boundaries are cycles but not all cycles are boundaries, and we have \(B_k({\mathcal M}) \subset Z_k({\mathcal M})\). The quotient space \({\mathcal H}_k({\mathcal M}) = [Z_k({\mathcal M})/B_k({\mathcal M})]\) is the homology spaces of order k of the mesh \({\mathcal M}\), and the Betti’s number \(b_k ={ \mathrm {rank}}\,[{\mathcal H}_k({\mathcal M})]\). The presence of curl-free fields (resp. div-free fields) that are not the gradient of a scalar field (resp. the curl of a vector field) is indicated from the fact that b 1≠0 (resp. b 2 ≠ 0). We recall that Betti’s numbers are topological invariants (i.e., they depend on the domain Ω up to a homeomorphism) and do not depend on the mesh \({\mathcal M}\) on \(\bar \varOmega \) that is used to compute them (see [14] and an application in [13]).
For the high order case, we need to introduce some concepts of relative homology. Let \(\mathcal K_k({\mathcal M})\) be subspaces of \(\mathcal C_k({\mathcal M})\) with \(\partial _k \mathcal K_k({\mathcal M}) \subset \mathcal K_{k-1}({\mathcal M})\). We thus say that \(c \in {\mathcal C}_k({\mathcal M})\) is closed [modulo \(\mathcal K_k({\mathcal M})\)] if \(\partial c \in \mathcal K_k({\mathcal M})\). A (k − 1)-chain c bounds [modulo \(\mathcal K_k({\mathcal M})\)] if there exists a k-chain γ such that \(c - \partial \gamma \in \mathcal K_{k-1}(\mathcal M)\). We thus talk about relative homology groups.
A k-cochain w (over the mesh \({\mathcal M}\)) is a linear mapping from \({\mathcal C}_k({\mathcal M})\) to \(\mathbb {R}\). They are discrete analogues to differential forms. For k > 0, the exterior derivative of the (k − 1)-form w is the k-form dw such that ∫sdw =∫∂s w for all \(s \in \mathcal {C}_{k} ({\mathcal M}).\, \) With this simple equation relating the evaluation of dw on a simplex s to the evaluation of w on the boundary of this simplex, the exterior derivative is readily defined. We can naturally extend the notion of evaluation of a differential form w on an arbitrary chain by linearity: \( \int _{\sum _i {\mathbf {c}}_i s_i} w = \sum _i {\mathbf {c}}_i \int _{s_i} w.\) Thus
The operator d is the dual of the boundary operator ∂. As a corollary of the boundary operator property ∂∂ = 0, we have that dd = 0. Since we used arrays of dimension \(|\varDelta _k({\mathcal M})|\) to represent a k-cochain, the operator d can be represented by a matrix d of dimension \(|\varDelta _k({\mathcal M})|\times |\varDelta _{k-1}({\mathcal M})|\), 1 ≤ k ≤ 3. Again, we have one matrix for the exterior derivative operator for each simplex dimension. When a metric is introduced on the ambient affine space, the exterior derivative operator d stands for grad, curl, div, according to the value of k from 1 to 3, and it is represented by, respectively, G, R, D, the connectivity matrices of the mesh \({\mathcal M}\).
3 Small Simplices, Weights and Potentials
We introduce the multi-index α = (α 0, …, α s) of s + 1 integers α i ≥ 0 and weight \(|\boldsymbol \alpha |=\sum _{i=1}^{s} \alpha _i\). The set of multi-indices α with s + 1 components and weight r is denoted \(\mathcal I(s+1, r)\). We denote by v i the (Cartesian) coordinates of the node n i in \(\mathbb R^3\). Given a multi-index \(\boldsymbol \alpha \in \mathcal I (4,r)\), and a k-subsimplex S of T, the small simplex {α, S} is the k-simplex that belongs to the small tetrahedron with barycenter at the point of coordinates \(\sum _{i=0}^3 [ (\frac {1}{4}+ \alpha _i){\mathbf {v}}_{{\sigma ^0_T(i)}}]/(r+1)\), which is parallel and 1∕(r + 1)-homothetic to the (big) sub-simplex S of T. The notation {α, S} was first defined in [7]. The set of small tetrahedra of order r + 1 > 1 can be visualized starting from the principal lattice L r+1(T) in the simplex \(T=\{n_{\sigma ^0_T(0)}\, n_{\sigma ^0_T(1)}\,n_{\sigma ^0_T(2)}\,n_{\sigma ^0_T(3)}\}\) defined as
and connecting its points by edges parallel to those of T. (See, e.g., Fig. 1.)
We denote by Λ k(Ω) the space of all smooth differential k-forms on Ω. The completion of Λ k(Ω) in the corresponding norm defines the Hilbert space L 2 Λ k(Ω). Let \( \mathcal P^-_{r+1} \varLambda ^k(T)\) be the space of so-called trimmed polynomial k-forms of degree r + 1 on T, with r ≥ 0, (as in [9]), and we define
where HΛ k(Ω) = {ω ∈ Λ k(Ω) : dω ∈ Λ k(Ω)} is a Hilbert space (see [5]).
Definition 1
The weights of a polynomial k-form \(u \in \mathcal P^-_{r+1} \varLambda ^k(T)\), with 0 ≤ k ≤ 3 and r ≥ 0, are the scalar quantities
on the small simplices {α, S} with \(\boldsymbol \alpha \in \mathcal I (4,r)\) and S ∈ Δ k(T).
We now list some remarkable properties of the small simplices which are useful in the tree construction.
Property 1
The weights (1) of a Whitney k-form \(u \in \mathcal P_{r+1}^- \varLambda ^k(T)\) on all the small simplex {α, S} of T are unisolvent, as stated in [9, Proposition 3.14]. The small simplices can thus support the degrees of freedom for fields \(u \in \mathcal P^-_{r+1} \varLambda ^k(T)\), with 0 ≤ k ≤ 3 and r ≥ 0. Since the result on unisolvence holds true also by replacing T with F ∈ Δ n−1(T) then \(\mbox{Tr}_F u \in {\mathcal P}_{r+1}^- \varLambda ^k(F)\) is uniquely determined by the weights on small simplices in F. It thus follows that a locally defined u, with \(u_{|T} \in \mathcal P_{r+1}^- \varLambda ^k(T)\) and single-valued weights, is in HΛ k(Ω). We thus can use the weights on the small simplices {α, S} as degrees of freedom for the fields in the finite element space \(\mathcal P_{r+1}^- \varLambda ^k({\mathcal M})\) being aware that their number is greater than the dimension of the space.
Property 2
The weights given in Definition 1 have a meaning as cochains and this relates directly the matrix describing the exterior derivative with the matrix of the boundary operator. The key point is the Stokes’ theorem ∫Cdu =∫∂C u , where u is a (k − 1)-form and C a k-chain. More precisely, if \(u \in \mathcal P_{r+1}^- \varLambda ^k({\mathcal M})\) then \(z = \mathrm {d} u \in \mathcal P_{r+1}^- \varLambda ^{k+1}({\mathcal M})\) and
being B the boundary matrix with as many rows as small simplices of dimension k and as many columns as small simplices of dimension k − 1. The small simplices {α, S} inherit the orientation of the simplex S so the coefficient B {α,S},{β,F} is equal to the coefficient B S,F of the boundary of the simplex S if β = α. This is straightforward if \( \mathop {\mbox{dim}}(F)>0\) and when \( \mathop {\mbox{dim}}(F)=0\), providing that small nodes in T are given in the notation {α, n} according to their position in the small simplices when fragmented (see Fig. 1 in [1]).
Property 3
The generated \((\mbox{ }^{r+2}_{\ \ 2})\) small faces on each face F of T, pave F together with the \((\mbox{ }^{r+1}_{\ \ 2})\) reversed triangles, denoted by ∇, contained in F. Similarly, the generated \((\mbox{ }^{r+3}_{\ \ 3})\) small tetrahedra contained in T pave T together with the \((\mbox{ }^{r+2}_{\ \ 3})\) octahedra, denoted by O, and the \((\mbox{ }^{r+1}_{\ \ 3})\) reversed tetrahedra, denoted by ⊥, contained in T, as shown in Fig. 1. Reversed octahedra and reversed tetrahedra are examples of “holes” in T (see [7, 12]).
Property 4
Since homology is preserved by homotopy, in [12, Section 3.4], it is discussed the fact that the relative homology (i.e., the homology [modulo the holes’ boundaries]), of the complex of small simplices is the same of the homology of \({\mathcal M}\). This property is fundamental to build the tree for high order potentials when working with small simplices. The homology [modulo the holes’ boundaries] can be translated in matrix notation, by showing that the boundary matrices associated with the small simplices, “modified” and “completed” (in a sense that we explain in the next section) by the relations [12, Proposition 3.5] are incidence matrices of a graph. To apply the theory presented in [12, Section 3.4] in a tetrahedron \(T \in \varDelta _3({\mathcal M})\), we need to introduce, for r > 0, two sets \({\mathcal K}_1\) and \({\mathcal K}_2\) of chains generated by the small simplices that belong to the boundary of some hole in T as follows:
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\({\mathcal K}_1\) are the chains generated by the boundary of the \((\mbox{ }^{r+1}_{\ \ 2})\) reversed triangle ∇⊂ F and that for each F ∈ Δ 2(T), and the boundary of the three faces out of four on the boundary ∂⊥ of each of the \((\mbox{ }^{r}_{2})\) reversed tetrahedra ⊥ in T;
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\({\mathcal K}_2\) are the chains generated by 4 out of 8 faces of the \((\mbox{ }^{r+2}_{\ \ 3})\) octahedra O in T. The involved faces are the small faces belonging to the boundary ∂O privated of ∂O ∩ (Δ 2(T) ∪ ∂⊥).
The two sets \({\mathcal K}_1\) and \({\mathcal K}_2\) satisfy the property \(\partial {\mathcal K}_2 \subset {\mathcal K}_1\), decisive to conclude that the relative homology [modulo the holes’ boundaries] of the complex of the small simplices is the same as the homology of the original mesh \({\mathcal M}\) [12].
4 Trees and Graphs
As stated in [14], a directed graph \({\mathcal G}\) consists of two sets \({\mathcal N}\) and \({\mathcal A}\) of nodes and arcs, respectively, subjected to certain incidence relations, collected in the all-vertex incidence matrix \(\mathtt {M}^{\mathcal G} \in {\mathbb Z}^{ |{\mathcal N}| \times |{\mathcal A}|}\) as follows:
An incidence matrix M of the graph \({\mathcal G}\) is any sub-matrix of \(\mathtt {M}^{\mathcal G}\) with \(|{\mathcal N}|-1\) rows and \( |{\mathcal A}|\) columns. The node that corresponds to the row of \(\mathtt {M}^{\mathcal G}\) that is not in M will be indicated as the reference node of \({\mathcal G}\). A graph \({\mathcal G}\) is connected if there is a path between any two of its nodes. A tree \({\mathcal T}\) of a graph \({\mathcal G}\) is a connected acyclic subgraph of \({\mathcal G}\). A spanning tree \({\mathcal T}_s\) is a tree of \({\mathcal G}\) visiting all its nodes. Any connected graph \({\mathcal G}\) admits a spanning tree \({\mathcal T}_s\). We have now to particularize these notions for small simplices. In each tetrahedron T of the oriented mesh \({\mathcal M}\), we consider the small mesh associated with L r+1(T) composed only of small tetrahedra, for a given r uniform all over the mesh \({\mathcal M}\). The union of the small meshes for all \(T \in \varDelta _3({\mathcal M})\) is denoted \({\mathcal M}_{all}\).
A (Primal) Small Tree for the Gradient Problem
For r = 0, the graph \({\mathcal G}^1\) has \({\mathcal N} = \varDelta _0({\mathcal M})\) and \({\mathcal A} = \varDelta _1({\mathcal M})\). The boundary matrix G ⊤ is the all-vertex incidence matrix of the graph \({\mathcal G}^1\). Estracting a spanning 1-tree \({\mathcal T}^1_s\) from \({\mathcal G}^1\) is equivalent to finding in G ⊤, minus one row, a submatrix of maximal rank (see [13] for a suitable and easy way of constructing \({\mathcal T}\)). For r > 0, we have to consider the new graph \({\mathcal G}^1\) with \({\mathcal N} = \varDelta _0({\mathcal M}_{all})\) and \({\mathcal A} = \varDelta _1({\mathcal M}_{all})\). Let \(\mathtt {G}_{all} ^\top \) be the all-vertex incidence matrix of this new graph \({\mathcal G}^1\). Note that \(\mathtt {G}_{all} ^\top \) results from the boundary operator ∂ 1 on the elementary 1-chains from \({\mathcal M}_{all}\). Estracting a spanning 1-tree \({\mathcal T}^1_s\) from \({\mathcal G}^1\) is equivalent to finding in \(\mathtt {G}_{all} ^\top \), minus one row, a submatrix of maximal rank. Example of spanning 1-tree \({\mathcal T}^1_s\) for r + 1 = 2 in the right part of Fig. 2 and for r + 1 = 3 in Fig. 5 (fragmented visualization). Note that we can repeat this construction in the two-dimensional case.
A (Dual) Small Tree for the Divergence Problem
For r = 0, the graph \({\mathcal G}^2\) is built on \({\mathcal M}^*\), the so-called dual mesh of \({\mathcal M}\), as follows. Let us note that an internal face \(F \in \varDelta _2({\mathcal M})\) connects two adjacent tetrahedra \(T_1,T_2 \in \varDelta _3({\mathcal M})\) whereas a boundary face \(F_b \in \varDelta _2({\mathcal M})\) connects a tetrahedron \(T_b \in \varDelta _3({\mathcal M})\) and the boundary ∂Ω. We can construct the following connected (dual) graph \({\mathcal G}^2\): the set of nodes, \({\mathcal N} \), contains the barycenter of any tetrahedron \(T \in \varDelta _3({\mathcal M})\) together with one additional exterior node representing ∂Ω; the set of arcs, \({\mathcal A}\), contains any face \(F \in \varDelta _2({\mathcal M})\). For r = 0, the matrix D associated with the boundary operator ∂ 3, acting on \(C_3({\mathcal M})\), is an incidence matrix of the (dual) graph \({\mathcal G}^2\), with reference node the one corresponding to ∂Ω. Estracting a spanning tree \({\mathcal T}^2_s\) from \({\mathcal G}^2\) is equivalent to finding in D a submatrix of maximal rank.
For r > 0, let \({\mathcal R}_2\) be the set of small faces chosen as follows: one small face for each octahedron O contained in \({\mathcal K}_2\) (see the right side of Fig. 3 for the dashed small face in \({\mathcal R}_2\) when r + 1 = 2). To construct the graph \({\mathcal G}^2\) for r > 0 we need to consider \({\mathcal M}^*_{all}\), the dual mesh associated to \({\mathcal M}_{all}\), where nodes are the small tetrahedra and the arcs the small faces, apart from the ones in \({\mathcal R}_2\). To understand this, we can reason as follows. For r > 0, we have one arc connecting two small tetrahedra, say t ◇, t ∘, when
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either t ◇, t ∘ share the same small face f, i.e. ∂t ◇∩ ∂t ∘ = f;
-
or t ◇, t ∘ have a small face on the boundary of the same octahedron O, i.e. f ◇ = ∂t ◇∩ ∂O and f ∘ = ∂t ∘∩ ∂O for the same octahedron O.
See an example of graph \({\mathcal G}^2\) for \({\mathcal M}_{all}\) (here \({\mathcal M} = \{T\}\)) in the left part of Fig. 3 for r + 1 = 2, where the node associated with the octahedron O is not a node in the graph, but stands to indicate that the four small tetrahedra are connected one to the other by one arc because they all have one small face on ∂O. Naming t k the small tetra with a vertex in v k, k = 0, 3, and numbering first the 3 × 4 faces on t k ∩ ∂T, called \(f^k_i\) for i = 1, 2, 3, second those on ∂O (where \(f^O_u\), \(f^O_\ell , f^O_d, f^O_r \) are the small faces up, left, down, right of ∂O), we have
Since the octahedron O is not part of the small mesh \({\mathcal M}_{all}\), we have to imagine that its node collapses with the node of one of its neighbouring small tetrahedron, say t 0 with a vertex in v 0, and thus that the corresponding arc (i.e. the small face \(f^O_{u}= \partial t_0 \cap \partial O\), the dashed one in the right part of Fig. 3) is eliminated. From a matrix point of view, D is obtained by adding the line “O” in D tmp to the line “t 0”, and eliminating \(f^O_{u}\), namely
(in bold font, the submatrix of maximal rank in D for the spanning tree \({\mathcal T}^2_s\) illustrated in Fig. 4, left part for r + 1 = 2). To repeat this construction in the two-dimensional case, when T is a triangle, we have to consider the mesh \({\mathcal M}_{all}\) of small triangles in T and the role of the core octahedra O is played by the reversed triangles ∇∈ T. The set \({\mathcal R}_2\) is replaced by \({\mathcal R}_1\), composed of one small edge for each reversed triangle \(\nabla \in {\mathcal K}_1\). In two dimensions we do not have reversed tetrahedra, therefore no reversed triangles ∇⊥.
The construction of the spanning tree in \({\mathcal M}_{all}\) can be done by assembling that of the geometrical mesh \(\mathcal M\), namely a spanning tree for the Whitney forms of lower degree (blue lines in Fig. 5 (Right)), together with local contributions, one from each element (green lines in Fig. 5 (Right)). Each local contribution results from one fixed on a reference element which is mapped on the current element (respecting the orientation). In Fig. 5 (Left), in green/red thick line we have marked the small edges of a spanning tree in the graph \({\mathcal G}^1\), for r = 3, in the reference triangle. The red ones belong to the spanning tree in the reference triangle, but they are in general omitted in the spanning tree of \({\mathcal M}_{all}\), (indeed, they appear only if they are covered by the blue tree). The small co-tree is in black. A similar construction can be repeated in 3D (both for k = 1 and k = 2) and it reflects the decomposition given, for instance, in [15] (Sect. 5).
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Rodríguez, A.A., Rapetti, F. (2020). Small Trees for High Order Whitney Elements. In: Sherwin, S.J., Moxey, D., Peiró, J., Vincent, P.E., Schwab, C. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018. Lecture Notes in Computational Science and Engineering, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-030-39647-3_47
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