Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
-
Carsten Carstensen
und
Jun Hu
Abstract
The global arrangement of the degrees of freedom in a standard Argyris finite element method (FEM) enforces
1 Introduction
1.1 Motivation
One of the most well-known conforming plate elements dates back to Argyris
[4, 8, 15]
with a quintic polynomial space
Adaptive mesh-refinement algorithms are well established with optimal convergence rates [2, 10, 12, 13, 24] merely for the Morley (cf. [11, 20] and the references therein) amongst the aforementioned finite element methods (FEM) for fourth-order problems. The a posteriori error control for the Argyris FEM is known for more than two decades and even included in the first book [26] on a posteriori error analysis. So it appears surprising that the important task is open to design a rate-optimal adaptive mesh-refinement algorithm and merely plain convergence has been achieved [23, Problem 44].
A closer look at the mathematical foundation of adaptive mesh-refinement reveals that the Argyris FEM leads to non-nestedness under mesh-refinements and this causes mathematical difficulties in the overall analysis in [2, 10, 12, 13, 24]. In fact, according to the knowledge of the authors, the only positive convergence result in the context of adaptive algorithms of conforming FEM that overcomes non-nestedness is [31] for a particular mesh-refinement strategy.
The treatment of non-nested subspaces has a longer tradition in the mathematical foundation of the multilevel solver, but uniform convergence of standard multigrid V-cycle algorithms has only been established for the Bogner–Fox–Schmit and the Powell–Sabin FEMs [30, 25]. Those are the conforming FEMs for fourth-order problems with the nestedness property under uniform mesh-refinement. See [28] for the Morley based preconditioner for the Argyris element for biharmonic equations.
The two examples at least indicate severe difficulties in the mathematical investigation of non-nested finite element spaces and this paper enforces the nestedness by a marginal extension of the global degrees of freedom.
1.2 Hierarchical Argyris FEM
The adaptive mesh-refinement algorithms
and the multigrid methods are typically based on admissible triangulations obtained by a mesh-refinement strategy such as the newest-vertex bisection
[24]. Given an initial triangulation
as the sum of all preceding finite element spaces: The nestedness of those sums
Suppose in addition that
(Herein
1.3 Contributions and Outline of This Paper
Section 2 gives all the details of the global scheme called hierarchical Argyris FEM because of one additional degree of freedom that is a one-sided second-order normal-normal derivative at a new interior node associated to the nodal basis function
The quasi-interpolation operator is one main tool in the a posteriori error analysis and in the split of multilevel schemes. Section 3 revisits [19] on the standard Argyris FEM and extends it to the discrete quasi-interpolation for the hierarchical version at hand in Theorem 2 with a discrete version that enables a proof of the discrete reliability from the axioms of adaptivity [10, 12] in Section 4. The second main result is the rate-optimality of an adaptive algorithm in Theorem 6 that in fact recovers the quartic convergence order of the scheme on polygons, even on non-convex ones, and clearly outperformed the aforementioned quadratic schemes in practice. Since there is no general convergence result available for the discontinuous Galerkin schemes, except for over-penalization as outlined in [3] for second-order problems, the presented adaptive hierarchical Argyris FEM is the most competitive numerical scheme for biharmonic equations and their relatives.
The third main result of this paper is the uniform convergence of a multigrid V-cycle algorithms with uniform convergence. Section 5 establishes a version for a sequence of successive one-level refinements based on newest-vertex bisection with local smoothing for the new nodes plus their neighbors. This is an effective algorithm known for second-order problems from [27] and Theorem 7 guarantees a bounded condition number for approximative inverse from the effective multigrid preconditioner for the biharmonic problem. The proofs in Sections 6 and 7 of this paper concern an auxiliary uniform triangulation that allow for standard approximation estimates from global interpolation of Sobolev norms. The link to the sequence of triangulations with local refinement utilized in the multigrid algorithm is through geometric arguments in [27] that on the specific geometric structure of bisection grids [14]. This paper replaces all geometry by a re-summation argument that is utterly algebraic and hence not restricted to 2D.
The analysis is outlined for the quintic version of the Argyris FEM for an easy and explicit presentation of the ideas. The results hold for the hierarchical conforming higher-order Argyris FEM as well. The motivating plate model problem is intrinsically 2D, and so the analysis is carried out in 2D, but the arguments does not rely on two space dimensions.
1.4 General Notation
Standard notation on Lebesgue and Sobolev
spaces and norms applies throughout this paper, e.g.,
The context-sensitive measure
Throughout this paper,
2 The Hierarchical Argyris FEM
Based on the class of admissible triangulations generated from successive mesh-refinement, the Argyris finite element
2.1 Admissible Triangulations
Given any initial triangulation
The bisection
Given any admissible triangulation
denote
the set of neighboring triangles that contain z. Let
denote the interior of the union of all neighboring triangles of T.
Within the definition of
which shall be relevant for the additional
global degrees of freedom in the extended Argyris finite element space
2.2 Extended Argyris Finite Element Space
The extended Argyris finite element space
The elementwise degrees of freedom of the Argyris finite element space are
not changed except for a transformation of another global coordinate system in the direction
The resulting global degrees of freedom are summarized for clarity for a given admissible triangulation
In this lexicographic order of multi-indices and derivatives
at z.
In this order those
Remark 2.1 (Dimension).
The dimension of the finite element space
for the number
Notation 2.2 (Nodes).
Given any admissible triangulation
Notation 2.3 (Degrees of Freedom).
There are
It is also convenient to use an alternative notation and to enumerate the set of degrees of freedom
Notation 2.4 (Nodal Basis).
Given the N global degrees of freedom
Remark 2.5 (Standard Argyris Finite Element).
The standard Argyris finite element space
is a linear subspace of
Unlike the classical Argyris finite element space
Theorem 1 (Nestedness).
Any triangulation
Proof.
It is straightforward to verify that any bisection of an interior edge
The general assertion follows from the above, because any NVB is obtained by a sequence of successive bisections of marked edges. ∎
3 Discrete Quasi-Interpolation
This section is devoted to the design of a quasi-interpolation operator for
the extended Argyris finite element spaces of this paper with properties (a)–(b) in this section as a tool for the proof of
discrete reliability in Section 4 and (c)–(d) for the multigrid analysis in Sections 6–7.
Given any triangle
Theorem 2 (Discrete Quasi-Interpolation).
Given the initial triangulation
It is important to understand quasi-interpolation not as a single
operator but rather as a class of operators with many choices [16] and the focus in this paper is on conditions from mesh-refinement.
There are two conditions (C1)–(C2) in the first part of this paper on the rate-optimal adaptive algorithm (and (C3)–(C4)
in the multigrid application in the second).
Recall that the extended Argyris finite element space
Condition (C1).
The family
for all
The next condition assures (a) for the admissible refinement
Condition (C2).
The family
and in addition, for all
If there is no coarse and fine triangle that contains z, i.e., if
Conditions (C1)–(C2) can always be fulfilled by some choice of
Definition of J in Theorem 2.
Given
Compared to the classical Argyris finite element space, the only modification
here is that some nodes have 7 rather than 6 degrees of freedom of this form. The shape functions are the same but the global arrangement
involves functionals with one-sided second-order normal-normal derivatives when either
is defined in terms of the nodal basis functions
for all
The two one-sided second-order normal-normal derivatives
The design of the weight functions implies
Proof of Theorem 2 (a).
Given
Since
The linear functionals
In case
at all nodes
Proof of Theorem 2 (b).
The arguments in the proof of [19, Theorem 7.1] show,
for any T with neighborhood
It is a standard technique [8] to combine a stability property with
an approximation property. If T has a positive distance to the boundary
with Poincaré inequalities in
This concludes the proof of (b). ∎
Proof of Theorem 2 (c).
The point is that Jf is well defined for
Proof of Theorem 2 (d).
Given any
If T has a positive distance to the boundary
If T hits the boundary
4 Adaptive Mesh-Refinement Algorithm
The axioms of adaptivity are four abstract properties (A1)–(A4) for an estimator η and a distance function δ to provide optimal convergence rates [10, 12, 6].
4.1 Error Estimator and Distance
For any admissible triangulation
with the scalar product a and the linear form F defined by
The unique discrete solution
Here and throughout the paper,
The analog formula applies to any admissible refinement
4.2 Stability (A1) and Reduction (A2)
The first index
The empty sum is zero and the full sum is abbreviated
Theorem 3 (Discrete Stability and Reduction).
There exist universal constants
4.3 Discrete Reliability (A3) and Quasi-Orthogonality (A4)
The discrete reliability is key to the a posteriori error analysis and follows with the discrete quasi-interpolation of Theorem 2 plus standard arguments.
Theorem 4 (Discrete Reliability).
There exists a universal constant
Proof.
Given the discrete solution
A piecewise integration by parts twice proves that the last residual term
The definition of
A (discrete) trace inequality and the approximation properties for
On the other hand, this contribution vanishes for all
The nestedness property of the extended Argyris finite element spaces implies orthogonality.
Theorem 5 (Quasi-Orthogonality).
Any sequence
with the universal constant
Proof.
Based on Theorem 1, the successive application of the Pythagoras theorem leads to
This and Theorem 4 imply, for all
The proof concludes with
4.4 Adaptive Algorithm and Optimal Convergence Rates
The subsequent adaptive mesh-refinement algorithm is the nowadays standard strategy.
Adaptive Algorithm.
The algorithm has the input parameter
compute
compute error estimator
mark
refine all
Output are the discrete solutions
Let
for the integral mean
The interpretation of the subsequent consequence is that of optimal convergence rates: If the right-hand side is smaller than infinity, then there is an optimal way to achieve the convergence rate s and the following theorem asserts that also the outcome of the adaptive algorithm converges with the rate s.
Theorem 6 (Optimal Rates).
There exists a universal constant
The constant
Proof.
This follows from the axioms of adaptivity [6, 10, 12] that (A1)–(A4)
and sufficiently small
This, the best-approximation property of the scheme,
and the equivalence of the error estimator plus data oscillations
5 Multigrid V-Cycle Algorithm
The first subsection introduces the overall V-cycle algorithm with some smoothing operator that requires a closer look at the degrees of freedom for the extended Argyris FEM in the second subsection.
5.1 The Multigrid V-Cycle Algorithm
Let
with the inner product
The prolongation and restriction operators are defined in terms of
the
for all
The local Gauss–Seidel relaxation of Section 5.2
defines a smoothing operator
The smoothing operator
Given an initial vector
with an approximate inverse
Algorithm (V-Cycle).
Let
Pre-smoothing:
Correction:
Post-smoothing:
Then
Theorem 7.
Under the assumptions of this subsection and with the
selection of the basis functions
for some positive
The proofs start after the description of the smoothing operator.
Theorem 7 implies that the simple iterative scheme
(5.5) converges with rate smaller than
or equal to some
5.2 Partial Gauss–Seidel Relaxation
The smoothing operator
The neighboring triangles
In other words, (5.6) is the absence of
for
Throughout the present paper, let
The nodal basis functions
The multigrid V-cycle algorithm is an adaptation of [27] (the Courant FEMs for second order problems) to the extended Argyris FEM. The multigrid V-cycle algorithm of [14] can also be generalized to the extended Argyris FEM.
6 Preliminaries in the Multigrid Analysis
This section starts with three preliminary subsections on the
6.1 Lumped Mass Matrix
Note that each degree of freedom
of their global numbers; recall that the triangle T is the compact
convex hull of its vertices and so its edges’ midpoints belong to T.
Any function
with the real coefficient
The extended as well as the classical Argyris finite elements are not affine equivalent
(although quasi-affine [15]) and
so the scaling and the lumped representation of the
Lemma 1 (Lumped
L
2
Norm).
Adopt the aforementioned notation for
Proof.
Any triangle
The degrees of freedom simply translate, but the scaling depends on
the order μ of the degree of freedom. Let the 21 degrees of freedom
with
To prove the lemma, suppose
holds for all
Since the scaling factor t satisfies
6.2 Re-summation
The analysis of multilevel methods requires global and local arguments. The re-summation lemma of this section links the locally refined triangulations from NVB with a structured uniform partition.
6.2.1 Three Constants in NVB
Given the initial triangulation
Some concepts are required to define
and recall
The second concept is the refinement level
Lemma 2 (Universal Bounds).
The three positive constants
exclusively depend on
Proof.
Recall that there are a finite number of different interior angles (at most
For each fixed triangle
6.2.2 Uniform Mesh-Refinement
Recall that the initial triangulation
defines a sequence of quasi-uniform admissible triangulation. Since
for the minimal resp. maximal initial mesh-size
6.2.3 Re-summation
Adopt the notation of the previous two subsections and consider
a sequence
for
For clarity
set
for any finite set
Lemma 3 (Re-summation).
Under the above notation,
holds for all
Proof.
Notice that the right-hand side is independent of L and so the upper index L in the sum
on the left-hand side could even be replaced by
The proof departs with the definition of
Consequently,
Any
and Lemma 2 asserts that there exist at most
The last identity utilizes that each admissible triangle T belongs to
for the multiplicity
of indices
Recall the sum convention
6.3 Analysis for Uniform Triangulations
All triangulations and the entire analysis throughout this subsection concern the uniform meshes
from Section 6.2.2
and so the upper index is skipped here (but added once the reference is made outside of this subsection),
so
The strengthened Cauchy–Schwarz inequality reads as follows.
Lemma 4.
Any
Proof.
The proof may follow the lines in [30, Lemma 5.1]; a simplified version is given
below based on the trace inequality and inverse estimates. Their combination is sometimes called discrete
trace inequality and asserts
The discrete trace inequality leads to
The square root of this, the above estimates, and inverse estimates verify
The sum of all those estimates over
Let
The quasi-uniform triangulations allow for approximation error estimates and for inverse estimates for
functions and norms in the Sobolev spaces
The following approximation and stability properties are well known for the quasi-uniform triangulations at hand in this subsection.
Lemma 5 ([30, Lemma 5.2]).
For any
hold for
The analysis requires the known reduced elliptic regularity on polygons that involves some
index
Theorem 8 (Regularity Shift).
Given the bounded Lipschitz domain
Proof.
The result has been established by Blum and Rannacher for convex domains with
The reduced elliptic regularity and the Aubin–Nitsche duality technique provide error estimates in weaker Sobolev norms; the proof is outlined for completeness.
Lemma 6 (Duality).
Any
Proof.
Given
conclude the proof with
Lemma 7 (Stability).
Any
Proof.
The proof follows that in [30, Lemma 5.3] and is merely outlined with less typos. On uniform meshes, the interpolation theory provides an inverse estimate
in fractional-order Sobolev spaces.
The stability in Lemma 5 (b) leads to
Lemma 8.
Any
Remark 6.1.
Lemma 8 (b) fails for
Proof of Lemma 8 (a).
The proof follows that of [30, Lemma 5.4].
Abbreviate
With the abbreviations
for the vector
Proof of Lemma 8 (b).
Set
After triangle inequalities this leads for
The remaining calculations follow the arguments of part (a) of the proof and utilize the last estimate to establish
The last sum is the product of the vector
7 Convergence Analysis
7.1 Overview
The proof of Theorem 7 concludes in Section 7.6 with arguments of a general multilevel theory [5, 29] based on certain decompositions. Throughout the analysis of the multigrid algorithm of Section 5, there is a given sequence
of an arbitrary
7.2 Quasi-Interpolation
The design of the quasi-interpolation operator in Theorem 2
looks ahead towards a refinement
Theorem 9.
Given a sequence
The definition of J is steered by a family
Condition (C3) is nothing else than a rewriting of (C1) for each level.
Condition (C3).
For any
Note that the set of nodes increases with the level
defines the space
Condition (C4).
For any
Definition of
(
J
ℓ
)
ℓ
∈
N
0
in Theorem 9.
Mathematical induction proves that a family
Proof of Theorem 9 (a’).
Consider
Consequently (with the duality in the first equality),
The inclusion in the last step follows from the above characterization of
7.3 Stability of the Decomposition
The subsequent proposition is of general interest where
Proposition 1 (Stability).
The decomposition (7.1) of
Proof.
The proof utilizes the
Step one provides for
The proof of (7.2) departs from the definition of
Hence suppose
For
Theorem 2 (d) also applies to each
Step two recalls the abbreviations
The proof of (7.4) departs with piecewise (with respect to
Recall that any
Since
Step three applies the re-summation of Lemma 3
with
for the left-hand side LHS in Proposition 1.
Step four finishes the proof. For any
This and Lemma 8.b for
7.4 Second Estimate for the Decomposition
The main argument for the following stability result requires Sobolev–Slobodeckij norms addressed before the proof concludes this subsection.
Proposition 2 (Key Estimate).
Any
The proof below requires Sobolev spaces
There are equivalent alternative definitions with equivalence constants that may depend on the domain ω. The decomposition
is obvious for the (non-local) Sobolev–Slobodeckij semi-norm.
Lemma 9 (Trace and Inverse Inequality).
Any
for a universal positive constant
Proof.
The trace and the inverse inequality are well established for a fixed domain such as the reference triangle
Proof of Proposition 2.
Adopt the above notation on
Step one establishes
Step two comments on the finite overlap of the supports of the nodal basis functions
Step three verifies that left-hand side LHS in Proposition 2 satisfies
Given any nodal basis function
Standard trace and inverse inequalities of the quintic polynomial
The sum is over all
The sum of the lower bounds for
Step four recalls the
for any admissible triangle
Step five establishes, for each
Any
provided
with
The combination with the previous estimate plus Theorem 2 (c)
with
This and
Step six concludes the proof. Steps two and four plus Lemma 3 show
Step four in the proof of Proposition 1 explains
Lemma 8.b applies to each of the three sums for
7.5 Third Estimate of a Decomposition
This subsection concerns an arbitrary discrete function
Recall that
Proposition 3 (Remains).
Any function
Proof.
Recall the notation from step two in the proof of Proposition 2
on the finite overlap of the supports of the basis functions
The equivalence of norms
The upper bound is a sum of
Consequently,
For a fixed
is followed by
Lemma 1.
Let
This is the identity (6.1) when
Replace
This and the aforementioned inverse inequality result in
for each
This concludes the proof. ∎
7.6 Proof of Theorem 7
The multigrid V-cycle operator
with adjoint
has been characterized in [29, Corolary 4.3] as
Given any
with
The bound of
The details of the proof of
can be written uniquely as a sum over
Abbreviate
The three sums in the upper bounds are named
Proof of
Proof of
The last identity follows from the definition of
Proof of
and the norm
This and Proposition 2 (note that
Finish of the proof. The summary of
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: Project-ID 255510958 – SPP 1748
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11625101
Funding statement: The first author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 255510958 – SPP 1748. The second author was supported by NSFC projects 11625101. The authors were partially supported by the Sino-German Center for Research Promotion under Grant GZ 1571.
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© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Sino–German Computational and Applied Mathematics
- An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
- Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
- Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
- On the Threshold Condition for Dörfler Marking
- An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
- Dimensionally Consistent Preconditioning for Saddle-Point Problems
- An Optimal Multilevel Method with One Smoothing Step for the Morley Element
- Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
- 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
- Defects in Active Nematics – Algorithms for Identification and Tracking
- Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
- Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
- Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
Artikel in diesem Heft
- Frontmatter
- Sino–German Computational and Applied Mathematics
- An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
- Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
- Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
- On the Threshold Condition for Dörfler Marking
- An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
- Dimensionally Consistent Preconditioning for Saddle-Point Problems
- An Optimal Multilevel Method with One Smoothing Step for the Morley Element
- Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
- 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
- Defects in Active Nematics – Algorithms for Identification and Tracking
- Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
- Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
- Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
