Multi-level hp-adaptivity and explicit error estimation
- Davide D’Angella^{1, 2}Email authorView ORCID ID profile,
- Nils Zander^{1},
- Stefan Kollmannsberger^{1},
- Felix Frischmann^{1},
- Ernst Rank^{1, 2},
- Andreas Schröder^{3} and
- Alessandro Reali^{2, 4}
https://doi.org/10.1186/s40323-016-0085-5
© The Author(s) 2016
Received: 11 July 2016
Accepted: 1 December 2016
Published: 30 December 2016
Abstract
Recently, a multi-level hp-version of the finite element method (FEM) was proposed to ease the difficulties of treating hanging nodes, while providing full hp-approximation capabilities. In the original paper, the refinement procedure made use of a-priori knowledge of the solution. However, adaptive procedures can produce discretizations which are more effective than an intuitive choice of element sizes h and polynomial degree distributions p. This is particularly prominent when a-priori knowledge of the solution is only vague or unavailable. The present contribution demonstrates that multi-level hp-adaptive schemes can be efficiently driven by an explicit a-posteriori error estimator. To this end, we adopt the classical residual-based error estimator. The main insight here is that its extension to multi-level hp-FEM is possible by considering the refined-most overlay elements as integration domains. We demonstrate on several two- and three-dimensional examples that exponential convergence rates can be obtained.
Keywords
Background
The finite element method (FEM) has been shown to produce particularly efficient approximations when both refinement in element size h and polynomial degree p are considered (hp-FEM). In this way, exponential convergence can be attained also in presence of singularities [1–3]. However, the implementation of hp-FEM is challenging. This is due to the fact that local refinements produce hanging nodes, edges and faces. The associated degrees of freedoms destroy the required \( C^0 \)-continuity of the approximations [4, 5] if not treated appropriately. To this end, it is a common approach to constrain the concerned degrees of freedom. However, it turns out that constraining becomes extremely complex, especially in cases where n-irregular meshes have to be handled in three dimensions. Thus, many codes only allow for 1-irregular meshes [3–5].
To overcome these difficulties, multi-level approaches have been developed, e.g., [6–12]. For a in-depth review, see, e.g., [13]. Along this line of research, the recently introduced multi-level hp-method allows to remove the hanging nodes by construction [14], while allowing the approximation capabilities comparable to the classical hp-FEM [15]. The underlying idea concerns the refinement procedure in which the refinement is not performed by replacement of elements, but by hierarchically overlaying a finer mesh. This technique is extensively explained in [13, 14] and briefly recaptured in “Multi-level FEM” section.
In previous contributions, the multi-level hp-refinement procedure made use of a-priori knowledge of the solution, e.g. singularities given by re-entrant corners. However, adaptive procedures can automatically produce discretizations that are more effective than intuitive choices of element sizes h and polynomial degree distributions p. This is particularly prominent when a-priori knowledge of the solution is only vague or unavailable.
The present contribution demonstrates that multi-level hp-adaptive schemes can be driven by an explicit a-posteriori error estimator as well. This kind of estimator was introduced, analyzed theoretically and used for adaptive computations in [16–20]. Successively, error estimation and adaptivity have become to be broadly researched and proven to be robust, see, e.g., [21–27]. Furthermore a first extension to high-order was given in, e.g, [18, 28, 29]. For a comprehensive survey, see, e.g., [30–34].
In the rest of the paper, “Estimated error-based hp-adaptivity” section briefly introduces the multi-level hp-method and the explicit error estimator. It is then demonstrated that the simple extension of viewing the refined-most elements as integration domains suffices for this classical method to drive multi-level adaptivity. Next, “Implementational aspects” section discusses some important implementational aspects. The article then proceeds with various numerical example in two- and three-dimensions in “Numerical examples” section.
Estimated error-based hp-adaptivity
Model problem
Multi-level FEM
Starting from a base mesh \({\mathcal {T}}^0(\Omega )=\lbrace T_{e^0} \rbrace _{e^0=1}^{n^0} \), additional meshes can be superimposed on domains of interest. Namely, one element \( T_{a} \in {\mathcal {T}}^0(\Omega ) \) can be refined by an overlay mesh \({\mathcal {T}}^1(T_{a})=\lbrace T_{a, e^1} \rbrace _{e^1=1}^{n^1}\). In the following, \( T_{a, e^1} \) will be called the sub-elements of \( T_{a} \), while \( T_{a} \) is the parent element of \( T_{a, e^1} \). Any element \( T_{a, b} \in {\mathcal {T}}^1(T_a) \) can be further refined by superimposing an additional mesh \( {\mathcal {T}}^2(T_{a, b}) \). For example, \( T_1\in {\mathcal {T}}^0 \) in Fig. 1a is overlaid by \( {\mathcal {T}}^1 = \lbrace T_{1,1}, T_{1,2} \rbrace \), while \( T_{1,1} \) is overlaid by \( {\mathcal {T}}^2=\lbrace T_{1,1,1}, T_{1,1,2}\rbrace \). This procedure of mesh-overlaying can be carried out an arbitrary number of times for each element of each overlay mesh. Finally, a polynomial degree has to be assigned to every element of each level to define its local basis. In this manner, a multi-level hierarchical structure of elements in defined.
The refinement-by-overlay procedure requires some precautions in order to ensure linear independence of the global shape functions and \( C^0 \)-continuity of the numerical solution. See [14] for details. For example, in one-dimensional domains \( C^0 \)-continuity can be guaranteed by requiring the local shape functions to vanish at the nodes of the underlying levels (c.f. Fig. 1a). Linear independence can be guaranteed by allowing high-order modes only on the leaf elements, i.e. the elements without any further refinement (c.f. Fig. 1a). Other polynomial degree distributions are possible, see [14] for details. The described multi-level mesh defines a set of global basis functions defined on the same domain \( \Omega \). For example, Fig. 1b shows the basis functions on \( \Omega =[0,1] \) from Fig. 1a collapsed to one level. This global basis can be used for classic analysis and a element-local basis point-of-view will be considered later in “Error estimator for multi-level FEM” and “Implementational aspects” sections.
Explicit error estimator
Many different techniques in error estimation can be found in the literature. A comprehensive survey is given in e.g. [30–34]. For the paper at hand, we follow the strategy of a-posteriori residual-based error estimators.
In general, the constant C is unknown and this makes it difficult to use Eq. (4) alone for assessing the quality of the numerical approximation. However, the element error indicators \( \eta ^2_{{{{T}}}} \) can identify the elements accounting for the highest error contributions. As illustrated later in “Smoothness indicator for the multi-level hp-adaptivity” section, this will be used to drive adaptivity.
Note that in [35, 36] Neumann boundary conditions were not considered. In the present paper, a direct heuristic generalization is formulated and investigated numerically.
Note also that it was proven for even degrees, that an explicit error estimator can be constructed just in terms of interior residuals [31, 38]. Instead, for odd degrees an explicit error estimator can be constructed using only the inter-element jumps [31, 39].
Error estimator for multi-level FEM
The first step towards using the error estimator (3) together with the multi-level FEM is to define what an element is. The estimator (3) employs a sum over all elements \( {T}\) of a traditional mesh \( {\mathcal {T}}\). However, as described above, multi-level meshes define a hierarchical structure of overlaid elements. Therefore, it is necessary to translate appropriately the multi-level structure to conventional finite elements. In this context, we will refer to the set \( {\mathcal {T}}\) to be used in (3) as the “partition” of the multi-level mesh \( {{\mathcal {T}}_{\textit{ml}}}\).
In the standard derivation of the explicit error estimator for the model problem, Green’s theorem is applied to express the energy norm of the approximation error in terms of interior- and boundary-residuals of each element [35, 37, 40]. This requires at least \( C^2 \)-continuity of the element local shape functions. In case of multi-level meshes, the local shape functions of a partition element \( {T}\) are the functions of all the levels \( {\mathcal {T}}^k_\textit{ml} \) that are non-zero on \( {T}\). Thus, not all choices for the partition of a multi-level mesh are suitable. For example, choosing \( \lbrace [0, 0.5] \), \( [0.5, 1] \rbrace \) as partition of \( {{\mathcal {T}}_{\textit{ml}}}\) in Fig. 1a only gives \( C^0 \)-continuity at \( x=0.25\).
It is noteworthy that this definition of partition as the traditional “elements” of the hierarchical multi-level mesh also complies with the set of sufficient conditions for the convergence of a finite element discretization given in [41]. Namely, for second-order problems the shape function shall represent exactly all polynomials of order up to one (completeness), be \( C^1 \)-continuous within each element (smoothness) and be \( C^0 \)-continuous across element boundaries (continuity). Despite the fact that these are just sufficient conditions, most of the finite element bases satisfy these requirements. Therefore, the above definition of elements of a multi-level mesh reveals to be meaningful in a more general sense, as it satisfies the smoothness requirement.
Note that, while each overlay mesh \( {\mathcal {T}}^k \) is assumed to be regular, \( {\mathcal {T}}^\textit{leaf}_{\textit{ml}}\) allows for an arbitrary level of hanging nodes (n-irregular mesh). These irregularities do not need complicated constraining algorithms, as \( C^0\)-continuity of the numerical solution is ensured by construction [14].
Smoothness indicator for the multi-level hp-adaptivity
In the context of adaptive refinement procedures, the error estimator allows to identify the elements that account for the highest error. Various methods exist to decide whether these elements should be p- or h-refined, see [2] for a comprehensive overview. In the sequel, the smoothness indicator proposed in [42] is employed. Its underlying idea is that for regions with a non-smooth solution, h-refinement is more effective than p-refinement, while areas with a smooth solution can be better approximated by p-refinement.
For one-dimensional problems, the Legendre coefficients of the numerical approximation \(u_{\mathcal {T}|{{T}}}\) local to \( {T}=[-1, 1] \) are used to indicate its smoothness. Denoting the i-th Legendre polynomial by \( P_i \), the Legendre coefficients can be computed as \( \alpha _i = (2 i + 1)/2 \int _{-1}^{1} P_i(x) u_{\mathcal {T}|{{T}}}(x) \; dx \), for \(0\le i \le p_{{{T}}}\). The decay rate \( \tau _{{T}}\) of these coefficients is estimated by a best fit of \( | \alpha _i | = c e^{\tau _{{T}}i} \). The decision between h- or p-refinement is then carried out depending on a parameter \( C_{\textit{decay}} \in \mathbb {R}\) as follows. If \( \tau _{{T}}\le C_{\textit{decay}} \), then \( u_{\mathcal {T}|{{T}}}\) is considered to be smooth and a p-refinement is performed. Otherwise, \( u_{\mathcal {T}|{{T}}}\) is classified as non-smooth and a h-refinement is chosen.
In the context of this work, only isotropic refinements are considered. In particular, p-refinements are performed if \( \tau _{{T}}=\max \lbrace \tau _{{{T}}x}, \tau _{{{T}}y} \rbrace \le C_{\textit{decay}} \). Otherwise, a multi-level h-refinement is carried out, as described below.
In the following, we consider as multi-level h-refinement of an element \( \bar{T}\), the superimposition on \( \bar{T}\) by by \( 2^d \) elements obtained by spatial bi-section of \( \bar{T}\) in all the d directions.
Implementational aspects
Differentiation recursive formulas
Evaluation of the flux jump
Consider an element \( T_1 \in {\mathcal {T}}^0_\textit{ml} \) of level 0 and its multi-level structure of refinements, as in Fig. 3. The key observation to evaluate the flux is that the ancestor \( \gamma _\textit{anc} \) of each boundary edge \( \gamma \) is internal to some element \( \tilde{T} \in {\mathcal {T}}^k_\textit{ml} \) of level k. Therefore, \( \gamma _\textit{anc} \) is contained in the volume of \( \tilde{T} \). As explained in “Error estimator for multi-level FEM” section, basis functions are assumed to be at least \( C^1 \)-continuous in the interior of \( \tilde{T} \). Hence, the shape functions defined locally on \( \tilde{T} \) do not give any contribution to the flux jump. Analogously, this holds for all the parent-elements of \( \tilde{T} \) of level \( l<k \). Therefore, it is in general sufficient to consider the contribution to the inter-element flux given by the sub-elements of \( \tilde{T} \) belonging to the level \( l>k \). For example, to evaluate the flux residual across AB, it is sufficient to evaluate the shape functions defined in \( T_{1,1} \) and \( T_{1,1,2} \) for the left flux. While to evaluate the right flux it is enough to consider the contribution given by \( T_{1,2} \), \( T_{1,2,1} \) and \( T_{1,2,1,1} \). In particular, no shape function needs to be evaluated in \( T_1\).
Numerical examples
L-shaped domain (2D)
Singular cube (3D)
To test adaptivity on three-dimensional singular solutions, the problem given by Eq. (1) is considered on \( \Omega =[0,1]^3 \) with \( \Gamma _N = \partial \Omega \), \( \Gamma _D = \left\{ \pmb {0} \right\} \). Moreover, \( f = \lambda (\lambda + 1) r^{\lambda -2} \) and \( g = \lambda r^{\lambda -2} \pmb {x} \cdot \pmb {n} \), where \( \lambda \in \mathbb {R}\), \( \lambda > 0 \), \( r(\pmb {x}) = \Vert \pmb {x}\Vert _2 \). The analytical solution \( u = r^\lambda \) has a gradient singular at \( \pmb {0} \) for \( \lambda < 2 \).
Shock cube (3D)
Geometrically complex example (3D)
Energy extrapolation values
p | #DOFs | Energy |
---|---|---|
18 | 379448 | 677.30107590527268257 |
19 | 445830 | 677.31338863697760643 |
20 | 519669 | 677.32359980041076141 |
Extrapolated energy | 677.39541370599670244 |
For this problem, no analytical solution is available such that the reference strain energy is approximated by the extrapolation described in [45, section 4.2]. The energies used for the extrapolation are obtained by uniform p-extensions using the trunk space [45] and listed in Table 1. For this type of geometry with edge singularities, an optimal hp-adaptive scheme is expected to achieve exponential convergence of the form (16) with \( \phi =1/4 \) [46]. This, however, requires anisotropic h-refinement which is still a subject of further research in the context of the multi-level FEM. An exponential decay of the error can, thus, not be expected. Nevertheless, the error estimator tracks closely the behavior of the error and the adaptive procedure drastically improves the convergence rate of the standard p-extension, as clearly shown in Fig. 12a–c.
Conclusions
This work demonstrates that the multi-level hp-adaptive scheme can be driven by means of an explicit error estimator and provides reasoning as well as the necessary formulae for its implementation. In this context, the decision between h- or p-refinement is carried out according to the decay rate of the Legendre coefficients of the numerical solution.
The considered examples include singularities or concentrated steep gradients which the error estimator tracks closely; the effectivity index also shows a converging behavior. Moreover, exponential convergence rates of the multi-level hp-adaptive procedure are shown. However, while the smoothness indicator performs excellently in the benchmark with simple geometries, this was not observed for the example presenting with a complex geometry. Nevertheless, efficient discretizations were found automatically also in this example.
Declarations
Author's contributions
All authors have prepared the manuscript. All authors read and approved the final manuscript.
Acknowledgements
With the support of the Technische Universität München-Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under Grant Agreement Number 291763. The authors also gratefully acknowledge the financial support of the German Research Foundation (DFG) under Grants RA 624/22-1 and RA 624/27-1, and Project SCHR 1244/4-1 of the priority programme SPP 1748.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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