Abstract
The computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate’s vibrations. This paper shows that the eigenvalue provided by the nonconforming Morley finite element analysis, which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense, is not always a lower bound. A fully-explicit error analysis of the Morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound. This paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower-order terms.
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1 Introduction
The Morley nonconforming finite element method provides asymptotic lower eigenvalue bounds for the problem \({\varDelta }^2 u = \lambda u\). It is observed in the numerical examples [8, p.39] that the Morley eigenvalue \(\lambda _{\mathrm{M}}\) is a lower bound of \(\lambda \). The possible conjecture that this is always the case, however, is false in general. This motivates the task to compute a guaranteed lower eigenvalue bound for all and even the very coarse triangulations based on the Morley finite element discretisation. This paper provides a guaranteed lower bound
for a computable value of \(\varepsilon \) which depends on the maximal mesh-size \(H\) and the type of the lower-order term, e.g., \(\varepsilon = 0.2574\, H^2\) for the eigenvalue problem \({\varDelta }^2 u = \lambda u\).
Let \({\varOmega }\subset \mathbb{R }^2\) be a bounded Lipschitz domain with polygonal boundary \(\partial {\varOmega }\) and outer unit normal \(\nu \). The boundary is decomposed in clamped (\({\varGamma }_C\)), simply supported (\({\varGamma }_S\)), and free (\({\varGamma }_F\)) parts
such that \({\varGamma }_C\) and \({\varGamma }_C\!\cup \!{\varGamma }_S\) are closed sets. The vector space of admissible functions reads
Provided the boundary conditions are imposed in such a way that the only affine function in \(V\) is identically zero, \(V\cap P_1({\varOmega }) = \{0\}\), the space \(V\) equipped with the scalar product
is a Hilbert space (colon denotes the usual scalar product of \(2\times 2\) matrices) with energy norm \(|\!|\!|\cdot |\!|\!|:=a(\cdot ,\cdot )^{1/2}\). Given a scalar product \(b\) on \(V\) with norm \(||\cdot ||:=b(\cdot ,\cdot )^{1/2}\), the weak form of the biharmonic eigenvalue problem seeks eigenpairs \((\lambda ,u)\in \mathbb{R }\times V\) with \(||u ||= 1 \) and
For a regular triangulation \({{\fancyscript{T}}}\) of \({\varOmega }\) with vertices \({\fancyscript{N}}\) and edges \({{\fancyscript{E}}}\) suppose that the interior of each boundary edge is contained in one of the parts \({\varGamma }_C\), \({\varGamma }_S\), or \({\varGamma }_F\), and let the piecewise action of the operators \(\nabla \) and \(D^2\) be denoted by \(\nabla _\mathrm{NC }\) and \(D^2_\mathrm{NC }\). The space of piecewise polynomials of total (resp. partial) degree \(k\) reads \(P_k({{\fancyscript{T}}})\) (resp. \(Q_k({{\fancyscript{T}}})\)). The Morley finite element space [4] with respect to a regular triangulation \({{\fancyscript{T}}}\) of \({\varOmega }\) equals
The finite element formulation of (1.2) is based on the discrete scalar product
and some extension \(b_\mathrm{NC }\) of \(b\) to the space \(V+V_{\mathrm{M}}\) with norm \(||\cdot ||_\mathrm{NC }:=b_\mathrm{NC }(\cdot ,\cdot )^{1/2}\). It seeks eigenpairs \((\lambda _{\mathrm{M}},u_{\mathrm{M}})\in \mathbb{R }\times V_{\mathrm{M}}\) such that \(||u_{\mathrm{M}} ||_\mathrm{NC }= 1\) and
The a priori error analysis can be found in [8]. For conforming finite element discretisations, the Rayleigh-Ritz principle [5], e.g., for the first eigenvalue
immediately results in upper bounds for the eigenvalue \(\lambda \). In many cases it is observed that nonconforming finite element methods provide lower bounds for \(\lambda \) and the paper [10] proves that the eigenvalues of the Morley FEM converge asymptotically from below in the case \(b(\cdot ,\cdot )=(\cdot ,\cdot )_{L^2({\varOmega })}\). This paper provides a counterexample to the possible conjecture that \(\lambda _{\mathrm{M}}\) is always a lower bound for \(\lambda \) and provides the guaranteed lower bound (1.1) for a known mesh-size function \(\varepsilon \). The main result, Theorem 1, implies (1.1) for any regular triangulation \({{\fancyscript{T}}}\) with maximal mesh-size \(H\) and \(\varepsilon = 0.2574\, H^2\). Theorem 2 provides lower bounds for higher eigenvalues.
The main tool for the explicit determination of \(\varepsilon \) is the \(L^2\) error estimate for the Morley interpolation operator from Theorem 3, which also opens the door to guaranteed error control for the Morley finite element discretisation of the biharmonic problem \({\varDelta }^2 u = f\). In comparison with the profound numerical experiments in [8], the theoretical findings of this paper allow guaranteed lower eigenvalue bounds via some immediate postprocessing on coarse meshes with reasonable accuracy even for mediocre refinements.
The remaining parts of the paper are organised as follows. Section 2 discusses the mentioned counterexample and shows that the Morley eigenvalue \(\lambda _{\mathrm{M}}\) may be larger than \(\lambda \). Section 3 establishes lower bounds for eigenvalues based on abstract assumptions on the Morley interpolation operator \(I_{\mathrm{M}}\). Section 4 provides \(L^2\) error estimates for \(I_{\mathrm{M}}\) with explicit constants that enable the results of Sect. 3 for different fourth-order eigenvalue problems. Section 5 presents applications to vibrations and buckling of plates with numerical results for various boundary conditions in the spirit of [8].
Throughout this paper, standard notation on Lebesgue and Sobolev spaces and their norms and the \(L^2\) scalar product \((\cdot ,\cdot )_{L^2({\varOmega })}\) is employed. The integral mean is denoted by ; the dot (resp. colon) denotes the Euclidean scalar product of vectors (resp. matrices). The measure \(|\cdot |\) is context-sensitive and refers to the number of elements of some finite set or the length \(|E|\) of an edge \(E\) or the area \(|T|\) of some domain \(T\) and not just the modulus of a real number or the Euclidean length of a vector.
2 Counterexample
The following counterexample shows that the possible conjecture that the Morley FEM always provides lower bounds is wrong. On the coarse triangulation of the square domain \({\varOmega }:=(0,1)\times (0,1)\) from Fig. 1a, the discrete eigenvalue for clamped boundary conditions \(\partial {\varOmega } = {\varGamma }_C\) computed by the Morley FEM is \({\lambda _{\mathrm{M}}} = 1.859\times 10^3\). The discrete eigenvalue computed by conforming FEMs is an upper bound for any lower bound of \(\lambda \). A computation with the conforming Bogner-Fox-Schmit bicubic finite element method leads to the first eigenvalue \({\lambda _\mathrm BFS } = 1.367\times 10^3\) on the partition from Figure 1b. Hence, \(\lambda _{\mathrm{M}}\) cannot be a lower bound for \(\lambda \). Table 1 contains the values for finer meshes and shows the convergence behaviour. The results of the subsequent sections lead to the guaranteed lower eigenvalue bounds of Table 1.
3 Lower eigenvalue bounds
This section establishes lower bounds for eigenvalues. The main tool is the Morley interpolation operator \(I_{\mathrm{M}}:V\rightarrow V_{\mathrm{M}}\), which acts on any \(v\in V\) by
where, for any \(E\in {{\fancyscript{E}}}\), the unit normal vector \(\nu _E\) has some fixed orientation and the midpoint of \(E\) is denoted by \(\mathrm{mid }(E)\). For any triangle \(T\) and \(v\in H^2(T)\), an integration by parts proves the integral mean property for the second derivatives
With the \(L^2\) projection \(\Pi _0 : L^2({\varOmega })\rightarrow P_0({{\fancyscript{T}}})\), this results in the global identity
The main assumption for guaranteed lower eigenvalue bounds is the following approximation assumption for some \(\varepsilon > 0\) which depends only on the triangulation and the boundaries \({\varGamma }_C\), \({\varGamma }_S\), \({\varGamma }_F\). Suppose
(The proof of (A) follows in Sect. 4 for various boundary conditions.)
Theorem 1
(Guaranteed lower bound for the first eigenvalue) Under the assumption (A) with parameter \(0<\!\varepsilon \!<\infty \), the first eigenpair \((\lambda ,u)\!\in \!\mathbb{R }\!\times \! V\) of the biharmonic operator and its discrete Morley FEM approximation \(({\lambda _{\mathrm{M}}},{u_{\mathrm{M}}})\in \mathbb{R }\times V_{\mathrm{M}}\) satisfy
Proof
The Rayleigh-Ritz principle on the continuous level and the projection property (3.1) for the Morley interpolation operator yield with the Pythagoras theorem
The Rayleigh-Ritz principle in the discrete space \(V_{\mathrm{M}}\) implies
The Cauchy inequality plus \(||u ||=1\) prove
Hence, the binomial formula and the Young inequality reveal for any \(0<\delta \le 1\)
Equation (3.2) and (A) lead to
The choice \(\delta := \varepsilon ^2\lambda _{\mathrm{M}}/(1 + \varepsilon ^2\lambda _{\mathrm{M}})\) concludes the proof. \(\square \)
Theorem 2
(Guaranteed lower bounds for higher eigenvalues) Under the conditions of Theorem 1 and sufficiently fine mesh-size in the sense that
holds for the \(J\)-th eigenpair \((\lambda _J,u_J)\in \mathbb{R }\times V\) of the biharmonic operator, the discrete Morley FEM approximation \((\lambda _{{\mathrm{M}},J},u_{{\mathrm{M}},J})\in \mathbb{R }\times V_{\mathrm{M}}\) satisfies
Remark
Although the exact eigenvalue \(\lambda _J\) is not known, any upper bound (e.g., by conforming finite element methods) will give a lower bound for the critical mesh-size.
The proof of Theorem 2 employs the following criterion for the linear independence of the Morley interpolants of the first \(J\) eigenfunctions.
Lemma 1
Let \((u_1,\dots ,u_J)\in V^J\) be the \(b\)-orthonormal system of the first \(J\) eigenfunctions and suppose (A) with parameter \(\varepsilon <(\sqrt{1+J^{-1}} -1)/\sqrt{\lambda _J}\), then the Morley interpolants \( I_{\mathrm{M}} u_1,\dots ,I_{\mathrm{M}} u_J\) are linearly independent.
Proof
The assumption (A) plus the projection property (3.1) imply for all \(j=1,\dots ,J\) that
This and the orthonormality of the eigenfunctions plus the Cauchy inequality show
The condition \(\varepsilon <(\sqrt{1+J^{-1}} -1)/\sqrt{\lambda _J}\) is equivalent to
This and the Gershgorin theorem prove that all eigenvalues of the mass matrix
are positive. \(\square \)
Proof of Theorem 2
The Rayleigh-Ritz principle reads
where the minimum runs over all subspaces \(V_J \subset V_{\mathrm{M}}\) with dimension smaller than or equal to \(J\). Lemma 1 guarantees that the vectors \(I_{\mathrm{M}} u_1,\dots ,I_{\mathrm{M}} u_J\) are linearly independent. Hence, there exist real coefficients \(\xi _1,\dots ,\xi _J\) with \(\sum _{j=1}^J \xi _j^2 = 1\) such that the maximiser of the Rayleigh quotient in \(\mathrm{span }\{I_{\mathrm{M}} u_1,\dots ,I_{\mathrm{M}} u_J\}\) is equal to \(\sum _{j=1}^J \xi _j I_{\mathrm{M}} u_j\). Therefore, \(v:=\sum _{j=1}^J \xi _j u_j\) satisfies
The projection property (3.1) and the orthogonality of the eigenfunctions prove
This and (3.4) yield
This estimate replaces (3.2) in the case of the first eigenvalue. The remaining parts of the proof are identical to the proof of Theorem 1 and, hence, omitted here. \(\square \)
4 \(L^{2}\) Error estimate for the Morley interpolation
This section provides error estimates for the Morley interpolation operator with explicit constants to guarantee the approximation assumption (A) of Sect. 3. Let \(j_{1,1} = 3.8317059702\) be the first positive root of the Bessel function of the first kind [6]. The following theorem provides an explicit \(L^2\) interpolation error estimate of the Morley interpolation operator with the constants
Theorem 3
(Error estimate Morley interpolation) On any triangle \(T\) with diameter \(h_T:=\mathrm{diam }(T)\), each \(v\in H^2(T)\) and its Morley interpolation \(I_{\mathrm{M}} v\) satisfy
The proof of Theorem 3 is based on the following two lemmas.
Lemma 2
(Trace inequality with weights) Any function \(f\in H^1(T)\) on a triangle \(T\) with some edge \(E\in {{\fancyscript{E}}}(T)\) satisfies
Proof
Let \(P\) denote the vertex opposite to \(E\), such that \(T=\mathrm{conv }(E\cup \{P\})\). For any \(g\in W^{1,1}(T)\), an integration by parts leads to the trace identity
The estimate \(|x-P|\le h_T\), for \(x\in T\), yields for \(g=f^2\)
Cauchy and Young inequalities imply, for any \(\alpha > 0\), that
\(\square \)
Lemma 3
(Friedrichs-type inequality) On any real bounded interval \((a,b)\) it holds
Proof
The bilinear form
defines a scalar product on \(H^1_0 (a,b)\) such that \(\left( H^1_0 (a,b), \left<\cdot ,\cdot \right>\right) \) is a Hilbert space. For any \(f\in H^1_0 (a,b)\) and the quadratic polynomial \(p(x) := (x-a)(b-x)\), a straight-forward calculation results in
On the other hand, the Cauchy inequality with respect to the scalar product \(\left<\cdot ,\cdot \right>\) reads
The combination of (4.2)–(4.3) leads to
The maximum is attained for \(f=p\). \(\square \)
The proof of Theorem 3 makes use of the Crouzeix-Raviart interpolation operator \(I_\mathrm CR \) [1, 2]. For a triangle \(T\), the Crouzeix-Raviart interpolation \(I_\mathrm CR : H^1(T)\rightarrow P_1(T)\) acts on \(v\in H^1(T)\) through
and enjoys the integral mean property of the gradient
The following refinement of the results from [3] gives an \(L^2\) error estimate with the explicit constant \(\kappa _\mathrm CR \) from the beginning of this section.
Theorem 4
(\(L^2\) error estimate for Crouzeix-Raviart interpolation) For any \(v\in H^1(T)\) on a triangle \(T\) with \(h_T:=\mathrm{diam }(T)\) the Crouzeix-Raviart interpolation operator satisfies
Proof
Let \(T=\mathrm{conv }\{P_1,P_2,P_3\}\) with set of edges \(\{E_1,E_2,E_3\} = {{\fancyscript{E}}}(T)\), the barycentre \(M:=\mathrm{mid }(T)\) and the sub-triangles (see Fig. 2)
The function \(f:=v-I_\mathrm CR v\) satisfies, for any edge \(E\in {{\fancyscript{E}}}(T)\),
Let denote the integral mean on \(T\). The trace identity (4.1) plus the Cauchy inequality reveal for those sub-triangles
Let, without loss of generality, \(M=0\) and so \(\sum _{j,k=1}^3 P_j\cdot P_k = 0\). An explicit calculation with the local mass matrix \(|T|/12\,(1+\delta _{jk})_{j,k=1,2,3}\) reveals
Hence,
The Pythagoras theorem yields
The Poincaré inequality with constant \(j_{1,1}^{-1}\) from [6] plus (4.5) reveal
\(\square \)
Proof of Theorem 3
The triangle inequality reveals for \(g:=v-{I_{\mathrm{M}}} v\) that
For the first term, Theorem 4 provides the estimate
The integral mean property (4.4) of the gradient allows for a Poincaré inequality
with the first positive root \(j_{1,1} = 3.8317059702\) of the Bessel function of the first kind [6]. This controls the first term in (4.6) as
Let \(E\in {{\fancyscript{E}}}(T)\) denote the set of edges of \(T\) and let the function \(\psi _E\in P_1(T)\) be the Crouzeix-Raviart basis function which satisfies
The definition of \({I_\mathrm CR }\) and the property \(\int \limits _T\psi _E\psi _F\; dx =0\) for \(E\ne F\) prove for the second term in (4.6) that
Since \(g\in H^1_0(E)\) for all \(E\in {{\fancyscript{E}}}(T)\), Lemma 3 implies
By the trace inequality (Lemma 2), this is bounded by
The definition of \(I_{\mathrm{M}}\) implies \(\nabla {I_{\mathrm{M}}} v = {I_\mathrm CR }\nabla v\). Since \(\nabla g = \nabla v - {I_\mathrm CR }\nabla v\), the arguments from (4.7) show
The combination of the preceding four displayed estimates leads to
The upper bound attains its minimum at \(\alpha = 1 /\kappa _\mathrm CR \). Altogether, (4.6), (4.8) and (4.9) lead to
\(\square \)
5 Numerical results
This section provides numerical experiments for the eigenvalue problems
on convex and nonconvex domains under various boundary conditions.
5.1 Mathematical models
5.1.1 Vibrations of plates
The weak form of the problem \({\varDelta }^2 u = \lambda u\) seeks eigenvalues \(\lambda \) and the deflection \(u\in V\) such that
for the bilinear form \(b(\cdot ,\cdot ) := (\cdot ,\cdot )_{L^2({\varOmega })}\). Its Morley finite element discretisation seeks \((\lambda _{\mathrm{M}},u_{\mathrm{M}})\in \mathbb{R }\times V_{\mathrm{M}}\) such that
Theorems 1–3 establish the lower bound \(J\)-th eigenvalue
for maximal mesh-size \(H^2<\left( \sqrt{1+J^{-1}} -1\right) /(\kappa _{\mathrm{M}}\sqrt{\lambda _J})\) in case of \(J\ge 2\).
5.1.2 Buckling
The weak form of the buckling problem \({\varDelta }^2 u = \mu {\varDelta } u\) seeks a parameter \(\mu \) and the deflection \(u\in V\) such that
for the bilinear form \(b(\cdot ,\cdot ) := (\nabla \cdot ,\nabla \cdot )_{L^2({\varOmega })}\). This model describes the critical parameter \(\mu \) in a stability analysis of a buckling plate loaded with a load in the plate’s midsurface times \(\mu \) [9]. Its Morley finite element discretisation seeks \((\mu _{\mathrm{M}},u_{\mathrm{M}})\in \mathbb{R }\times V_{\mathrm{M}}\) such that
with the piecewise version \(b_\mathrm{NC }(\cdot ,\cdot ) := (\nabla _\mathrm{NC }\cdot ,\nabla _\mathrm{NC }\cdot )_{L^2({\varOmega })}\).
Theorems 1–3 establish the lower bound \(J\)-th eigenvalue
for maximal mesh-size \(H<\left( \sqrt{1+J^{-1}} -1\right) /(\kappa _\mathrm CR \sqrt{\mu _J})\) in case of \(J\ge 2\).
5.2 Domains and boundary conditions
The a priori error analysis of the Morley finite element method in [8] has been accompanied by various numerical examples which are easily recast into guaranteed lower bounds via the theoretical findings of this paper. The benchmark examples of this section also consider higher eigenvalues and nonconvex domains.
The domains under consideration are the unit square \({\varOmega }=(0,1)^2\) and the plate with hole \((0,1)^2\setminus ([0.35,0.65]^2)\). Figure 3 describes the boundary conditions for the unit square, while Fig. 4 shows the boundary conditions for the plate with hole. The different parts of the boundary \(\partial {\varOmega }\) are indicated by the following symbols.
5.3 Further remarks on numerical experiments
5.3.1 Numerical realisation
The first eigenvalues of (5.1) are approximated by the Morley FEM (Fig. 5a) on a sequence of successively red-refined triangulations (i.e., each triangle is split into four congruent sub-triangles) based on the initial triangulations of Fig. 6a.
For comparison, the discrete eigenvalues of the conforming Bogner-Fox-Schmit FEM (Fig. 5b) are computed as upper bounds. The conforming finite element space reads \(V_\mathrm BFS := V\cap Q_3({{\fancyscript{T}}})\) with the values of the function, its gradient and its mixed second derivative at the free vertices as degrees of freedom as displayed in Fig. 5b. The computations are based on the initial partitions of Fig. 6b.
5.3.2 Higher eigenvalues
To illustrate the result for higher eigenvalues, the tables in 5.4.3 display the approximations for the 20th eigenvalue on the unit square under the boundary conditions 3a and 3e. The required minimal mesh-size for the lower bound according to Theorem 2 leads to \(h<0.016\) (resp. \(0.017\)) for example 3a (resp. 3e), where the upper bounds \(\lambda _\mathrm{BFS ,20}\ge \lambda \) are used to guarantee a sufficiently fine mesh. This separation condition is satisfied for the last three values of GLB and, therefore, those are valid bounds. (The values in brackets are not necessarily reliable bounds.)
5.3.3 Inexact solve
The estimates from Section 3 are derived under the unrealistic assumption that the discrete algebraic eigenvalue problems are solved exactly. However, since the term \(\lambda _{\mathrm{M}}/(1+\varepsilon ^2)\lambda _{\mathrm{M}}\) is monotone in \(\lambda _{\mathrm{M}}\), any lower bound for the discrete eigenvalue \(\lambda _{\mathrm{M}}\) yields a lower bound for \(\lambda \). In this sense, this paper reduces the task of guaranteed lower bounds of the eigenvalue problem on the continuous level via the Morley discretisation and sharp interpolation error estimates to the task of guaranteed lower eigenvalue bounds of the algebraic eigenvalue problem in numerical linear algebra. There are many results available for the localisation of eigenvalues in the finite-dimensional algebraic eigenvalue problems in the literature, e.g., in [7]. Throughout this paper and the numerical examples of this section, all numbers provided are computed with the ARPACK and the default parameters.
5.4 Results
The tables display the eigenvalue of the Morley FEM and the guaranteed lower bound (GLB). The eigenvalue of the conforming Bogner-Fox-Schmit FEM is given as an upper bound for comparison. The dash indicates out of memory (8 million degrees of freedom).
5.4.1 First eigenvalue for \({\varDelta }^2 u = \lambda u\) on the unit square
\(\lambda _{\mathrm{M}}\) | GLB | \(\lambda _\mathrm{BFS }\) |
---|---|---|
(Boundary condition 3a) | ||
288.36704 | 222.04958 | 1,367.8580 |
637.14901 | 611.91175 | 1,300.1260 |
1,008.8296 | 1,004.7288 | 1,295.3400 |
1,205.7698 | 1,205.4022 | 1,294.9632 |
1,271.0486 | 1,271.0230 | 1,294.9359 |
1,288.8461 | 1,288.8444 | 1,294.9341 |
1,293.4041 | 1,293.4039 | 1,294.9340 |
1,294.5510 | 1,294.5509 | 1,294.9340 |
1,294.8381 | 1,294.8380 | 1,294.9336 |
(Boundary condition 3b) | ||
9.4115855 | 9.3207312 | 12.480192 |
11.429097 | 11.420647 | 12.374319 |
12.109523 | 12.108929 | 12.363172 |
12.297560 | 12.297521 | 12.362415 |
12.346044 | 12.346041 | 12.362367 |
12.358275 | 12.358274 | 12.362364 |
12.361341 | 12.361340 | 12.362363 |
12.362103 | 12.362102 | 12.362362 |
12.362252 | 12.362251 | 12.362339 |
(Boundary condition 3c) | ||
118.46317 | 105.51708 | 516.92308 |
269.41278 | 264.79492 | 501.89357 |
409.86191 | 409.18341 | 500.64841 |
474.00642 | 473.94961 | 500.56920 |
493.62006 | 493.61620 | 500.56423 |
498.80737 | 498.80712 | 500.56392 |
500.12344 | 500.12342 | 500.56390 |
500.45370 | 500.45369 | 500.56388 |
500.53630 | 500.53629 | 500.56352 |
(Boundary condition 3d) | ||
270.01217 | 211.00461 | 870.28523 |
486.48522 | 471.63317 | 840.23446 |
693.94950 | 692.00668 | 838.28577 |
794.82321 | 794.66350 | 838.16022 |
826.71488 | 826.70407 | 838.15227 |
835.25025 | 835.24956 | 838.15177 |
837.42356 | 837.42351 | 838.15174 |
837.96950 | 837.96949 | 838.15171 |
838.10612 | 838.10611 | 838.15136 |
(Boundary condition 3e) | ||
239.60730 | 191.96836 | 440.00000 |
323.40541 | 316.77395 | 391.31816 |
368.87652 | 368.32684 | 389.74036 |
384.07793 | 384.04063 | 389.64282 |
388.22057 | 388.21818 | 389.63677 |
389.28068 | 389.28053 | 389.63639 |
389.54733 | 389.54732 | 389.63637 |
389.61409 | 389.61408 | 389.63636 |
389.63075 | 389.63074 | 389.63634 |
5.4.2 First eigenvalue for \({\varDelta }^2 u = \lambda u\) on the square with hole
\(\lambda _{\mathrm{M}}\) | GLB | \(\lambda _\mathrm BFS \) |
---|---|---|
(Boundary condition 4a) | ||
6,605.7795 | 242.12417 | 31,270.769 |
7,555.1473 | 2,624.4604 | 28,314.668 |
15,294.185 | 12,356.931 | 27,458.216 |
21,971.980 | 21,512.833 | 27,138.816 |
25,144.874 | 25,106.547 | 27,005.952 |
26,279.553 | 26,276.932 | 26,947.608 |
26,665.404 | 26,665.235 | 26,921.219 |
26,803.927 | 26,803.916 | 26,909.085 |
26,857.825 | 26,857.824 | – |
(Boundary condition 4b) | ||
741.11343 | 187.68590 | 1,862.4481 |
1,246.1046 | 951.31959 | 1,855.3809 |
1,626.2648 | 1,586.1738 | 1,851.9814 |
1,784.0893 | 1,781.0028 | 1,850.9178 |
1,832.2899 | 1,832.0860 | 1,850.6129 |
1,845.5824 | 1,845.5694 | 1,850.5473 |
1,849.1916 | 1,849.1907 | 1,850.5479 |
1,850.1867 | 1,850.1866 | 1,850.5609 |
1,850.4690 | 1,850.4689 | 1,850.5712 |
5.4.3 Higher eigenvalues for \({\varDelta }^2 u = \lambda u\) on the square domain
(Boundary condition 3a) | \(H{\times }10^{-1}\) | (Boundary condition 3e) | ||||
---|---|---|---|---|---|---|
\(\lambda _{{\mathrm{M}},20}\) | GLB | \(\lambda _\mathrm{BFS ,20}\) | \(\lambda _{{\mathrm{M}},20}\) | GLB | \(\lambda _\mathrm{BFS ,20}\) | |
33,194.719 | (938.24364) | 180,927.73 | 35.35 | 16,884.905 | (913.30834) | 112,640.00 |
56,445.852 | (12,128.990) | 139,642.27 | 17.67 | 53,924.215 | (12,008.327) | 100,177.45 |
102,198.50 | (72,303.616) | 138,018.79 | 8.838 | 83,810.508 | (62,588.541) | 99,773.533 |
125,411.88 | (121,557.17) | 137,905.04 | 4.419 | 94,755.581 | (92,538.410) | 99,748.561 |
134,423.04 | (134,138.08) | 137,897.32 | 2.209 | 98,415.381 | (98,262.552) | 99,747.012 |
137,002.79 | 136,984.25 | 137,896.82 | 1.104 | 99,408.352 | 99,398.592 | 99,746.916 |
137,671.63 | 137,670.45 | 137,896.79 | 0.552 | 99,661.908 | 99,661.294 | 99,746.910 |
137,840.39 | 137,840.31 | 137,896.79 | 0.276 | 99,725.636 | 99,725.597 | 99,746.909 |
5.4.4 First eigenvalue for \({\varDelta }^2 u = \mu {\varDelta } u\) on the unit square
\(\mu _{\mathrm{M}}\) | GLB | \(\lambda _\mathrm BFS \) |
---|---|---|
(Boundary condition 3a) | ||
30.430781 | 22.737880 | 52.923077 |
46.100761 | 40.864499 | 52.576696 |
50.603228 | 48.884311 | 52.362578 |
51.874002 | 51.410714 | 52.345894 |
52.223278 | 52.105101 | 52.344768 |
52.314035 | 52.284337 | 52.344696 |
52.337005 | 52.329570 | 52.344691 |
52.342768 | 52.340908 | 52.344690 |
52.344208 | 52.343743 | 52.344676 |
(Boundary condition 3b) | ||
2.4064529 | 2.3437460 | 2.4859617 |
2.4518157 | 2.4352200 | 2.4686648 |
2.4634618 | 2.4592520 | 2.4674819 |
2.4664122 | 2.4653558 | 2.4674062 |
2.4671535 | 2.4668891 | 2.4674014 |
2.4673392 | 2.4672731 | 2.4674011 |
2.4673857 | 2.4673691 | 2.4674011 |
2.4673963 | 2.4673921 | 2.4674008 |
2.4673914 | 2.4673903 | 2.4673965 |
(Boundary condition 3c) | ||
13.403557 | 11.665198 | 33.066754 |
22.926106 | 21.552699 | 32.417350 |
29.338870 | 28.752692 | 32.293439 |
31.461504 | 31.290487 | 32.275273 |
32.056325 | 32.011758 | 32.272463 |
32.215866 | 32.204601 | 32.272026 |
32.257575 | 32.254750 | 32.271958 |
32.268298 | 32.267591 | 32.271947 |
32.271024 | 32.270847 | 32.271931 |
(Boundary condition 3d) | ||
24.000000 | 18.944891 | 38.176592 |
32.093424 | 29.465032 | 37.880015 |
36.168341 | 35.281625 | 37.805108 |
37.377276 | 37.136145 | 37.799957 |
37.692922 | 37.631319 | 37.799628 |
37.772852 | 37.757367 | 37.799607 |
37.792911 | 37.789034 | 37.799606 |
37.797931 | 37.796961 | 37.799604 |
37.799185 | 37.798942 | 37.799589 |
(Boundary condition 3e) | ||
18.334369 | 15.229883 | 22.000000 |
19.443160 | 18.446280 | 19.817243 |
19.667256 | 19.402101 | 19.744335 |
19.721247 | 19.653913 | 19.739533 |
19.734714 | 19.717814 | 19.739229 |
19.738084 | 19.733854 | 19.739210 |
19.738928 | 19.737870 | 19.739209 |
19.739138 | 19.738873 | 19.739209 |
19.739189 | 19.739122 | 19.739208 |
5.4.5 First eigenvalue of \({\varDelta }^2 u = \mu {\varDelta } u\) on the square with hole
\(\mu _{\mathrm{M}}\) | GLB | \(\mu _\mathrm BFS \) |
---|---|---|
(Boundary condition 4a) | ||
65.836950 | 27.041396 | 277.04748 |
140.00805 | 79.426405 | 265.59236 |
210.08856 | 163.34921 | 260.75573 |
239.26986 | 221.24527 | 258.83912 |
250.18631 | 244.96933 | 257.99165 |
254.32732 | 252.95825 | 257.60556 |
255.99152 | 255.64335 | 257.42742 |
256.69732 | 256.60970 | 257.34466 |
257.00896 | 256.98699 | – |
(Boundary condition 4b) | ||
31.637668 | 18.726875 | 43.732101 |
38.366123 | 31.733452 | 42.623655 |
40.936657 | 38.774808 | 42.387116 |
41.849116 | 41.261178 | 42.326353 |
42.155629 | 42.004899 | 42.311264 |
42.257614 | 42.219647 | 42.308505 |
42.292109 | 42.282595 | 42.308707 |
42.304027 | 42.301646 | 42.309345 |
42.308224 | 42.307628 | 42.309833 |
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This work was supported by the DFG Research Center MATHEON.
Dedicated to Dietrich Braess on the occasion of his 75th birthday.
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Carstensen, C., Gallistl, D. Guaranteed lower eigenvalue bounds for the biharmonic equation. Numer. Math. 126, 33–51 (2014). https://doi.org/10.1007/s00211-013-0559-z
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DOI: https://doi.org/10.1007/s00211-013-0559-z