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Adaptive Morley FEM for the von Kármán equations with optimal convergence rates. (English) Zbl 1467.65107

The paper builds upon the previous work of the first authors et al. [IMA J. Numer. Anal. 41, No. 1, 164–205 (2021; Zbl 1460.65145)] and relies on various published works by their collaborators and the research fraternity. This paper provides the first rate-optimal adaptive algorithm for the von Kármán equations, a semi-linear elliptic coupled system of two fourth-order partial differential equations with two independent spatial variables, serving as the mathematical model. The beautiful mathematical structure of the von Kármán equations has attracted a considerable amount of mathematical analysis, in the past two decades. Nonetheless, their physical soundness has been often seriously questioned. Many efforts have been done towards understanding the physical and mathematical mechanism of the von Kármán equations by different researchers using numerical simulations. Literature surveyed by the authors reveals that the approximation and error bounds for regular solutions to von Kármán plate model, using conforming, mixed, and hybrid FEMs and nonconforming FEMs, a \(C^0\) interior penalty method, and discontinuous Galerkin FEMs have been thoroughly investigated. However, little is known in the literature about adaptive FEMs and their convergence rates for semi-linear problems, motivating adaptive nonconforming Morley finite element approximation. The paper therefore, is devoted to the convergence and optimality analysis of the adaptive Morley FEM for the fourth order elliptic problem. The fact that the Morley element has the least number of degrees of freedom on each element for fourth order boundary value problems is well-known. By its very definition, it only consists of piecewise quadratic functions with respect to every triangle of a given triangulation. It is also known that the consistency error of the Morley element is the trickiest part and plays a key role in its total error estimate. Most importantly, nonconforming Morley FEM is parameter free. It may be noted that the main ideas and results presented here are motivated from [loc. cit; IMA J. Numer. Anal. 39, No. 1, 167–200 (2019; Zbl 1465.65129); C. Carstensen and S. Puttkammer, J. Comput. Math. 38, No. 1, 142–175 (2020; Zbl 1463.65362); the first author and H. Rabus, SIAM J. Numer. Anal. 55, No. 6, 2644–2665 (2017; Zbl 1377.65147)], and can be viewed as its extension to nonconforming finite element methods for fourth order problems.
Technical aspects of the paper
The first subsection begins with weak formulation of the mathematical model of von Kármán equations followed by boundedness and ellipticity properties of bilinear form to guarantee the existence and uniqueness of the weak solution of BVP under the assumptions of Lax-Milgram theorem. The regular solution is sought from the vectorization of the variational approach. The variational approach also give solid mathematical foundation and make the error analysis more systematic. Authors, in this paper, aim to obtain a local approximation to an arbitrary regular solution. In the next subsection, regular triangulation of the polygonal domain and admissible mesh-refinement strategies are discussed leading to discretization by constructing a finite dimensional subspace \(V_h\) based on triangulations of the domain. In this subsection, the Morley element is defined and some preliminaries are introduced for the theoretical framework. Piecewise polynomials of degree at most \(k\) and the oscillation of the load function are considered as the key ingredients. In the next subsection, authors introduce some interpolation and enhancement operators, which will play a key role in the new consistency error estimate of the Morley element. From earlier published works, some preliminary lemmas are recalled leading to priori error analysis for the Morley FEM. It is well understood that, optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity. Four main arguments compose the set of axioms and identify necessary conditions for optimal convergence of adaptive mesh-refining algorithms. This abstract framework answers questions like: What is the state-of-the-art technique for the design of an optimal adaptive mesh-refining strategy, and which ingredients are really necessary to guarantee quasi-optimal rates? The four axioms are sufficient for optimal convergence rate, asserting stability on non-refined elements stability on non-refined elements, reduction property on refined elements, quasi-orthogonality and discrete reliability and finally achieving the goal set by the authors. In the reminder of the paper, given the exact and the corresponding discrete solutions, a posteriori error is analysed, which proves that the adaptive algorithm will asymptotically recover the best optimal convergence rate. Summarising, the abstract framework of the paper is independent of the precise application and its respective discretization.
Observations and comments
1)
The paper focusses on the von Kármán equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semi-linear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation.
2)
Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strategies versus non-adaptive strategies.
3)
In order to foster the development of a convergence theory and improved design of AFEMs in computational engineering and sciences, this paper describes a particular version of an AFEM and analyses convergence results.
4)
The main results of this work state convergence and optimality of the adaptive algorithm in the sense that the error estimator converges with optimal convergence rate.
5)
Since the actual adaptive algorithm only knows the estimator, reliability estimates have been a crucial ingredient for convergence proofs of adaptive schemes of any kind.
6)
This work provides some unifying framework on the theory of adaptive algorithms and the related convergence and quasi-optimality analysis.
7)
The impact of adaptive mesh-refinement in computational partial differential equations (PDEs) cannot be overestimated.
8)
The so-called nonconforming finite element method which had and still has a great impact on the development of finite element methods.
9)
Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptively. One key ingredient is the discrete reliability of a residual-based a posteriori error estimator, which controls the error of two discrete finite element solutions based on two nested triangulations.
10)
The need for a rigorous treatment of both mathematical and algorithmic issues is emphasised and the practically important aspects of automatic data generation and adaptive mesh refinement procedures are discussed.
11)
An adaptive meshing method based on element error estimation followed by suitable mesh refinement is a promising methodology.
12)
Nonconforming FEMs avoid the strong restrictions of conforming FEMs. For example, relaxation of \(C^{m-1}\) continuity of the finite element space of conforming FEMs is one such.
13)
There are situations when the use of non-conforming FEM is a plus point. Non-conforming elements are reported to have been used successfully in the displacement analysis of plate bending.
14)
Throughout this paper, authors adopt the standard notations and conventions for Lebesgue and Sobolev norms and semi-norms of a function defined on an open set.
15)
Solely four axioms of adaptivity guarantee the optimality in terms of the error estimators.

Contributions of this work
1)
This paper provides the first rate-optimal adaptive algorithm for the von Kármán equations.
2)
This paper presents an adaptive algorithm AMFEM and establishes optimal convergence rates for all sufficiently small positive input and bulk parameters.
3)
This paper establishes the adaptive Morley FEM as the first scheme with guaranteed optimal rates for the von Kármán plate model. It is therefore suggested as the method of choice for a nonconvex domain.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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References:

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