We now discuss how to find an optimal shape parameter c. The main idea to finding an optimal c is to search a suitable interval that presumably contains the optimal shape parameter c∗. We first solve (2.8) with c = c1 in the interval [0, 2], say, by setting c1 = 0.1, 0.2, ..., 2.0 and locate the best c∗1 = 1.0, say, according to the error estimator EEE or RQEE. We next choose [0.9, 1.1] as the next interval to search for c∗2 = 1.1, say. Then choosing [1.09, 1.11], we find the next c∗3 and so on. In other words, the intervals are shrunk digit by digit until an optimal c is found within the maximum computer accuracy in MATLAB which is about 10−16.
Algorithm 2 Optimal c
(0) Set nodeL = a, nodeU = b for an interval [a, b] and Mindigit to be the minimum computer number or the user defined minimum. Set i = 1.
(1) Find an optimal parameter c∗i by varying ci from nodeL to nodeU with a digit-wise increment by using either EEE or RQEE.
(2) Create a new interval with the center being c∗i at the next digit level, set i = i + 1, and go to Step (0).
As mentioned before, we generally do not have exact solution for using EEE. RQEE is more appropriate for application. We now derive the formula of RQEE. We multiply equation (2.1) by an arbitrary test function v(x)
v(x)∈ H1(Ω) : =
½ v(x)
¯¯
¯¯ Z 1
0
h
v2 + (v0)2 i
dx <∞
¾
where H1(Ω) is a Sobolev function space such that any function of H1 and it’s first derivative are square integrable. Then integrating the equation on [0, 1], we have
− Z 1
0
u00vdx = λ Z 1
0
uvdx
Integration by parts yields
Since the test function v is an arbitrary function, we choose v such that v ∈ H01(Ω) = {v ∈ H1(Ω) : v(x) = 0 x∈ ∂Ω} . Then equation (2.8) becomes and hence we can use
eλ = R1
0(u0h)2dx R1
0 u2hdx (2.10)
in place of the exact eigenvalue λ to define RQEE =
°°
°eλ − λh
°°
°.
2.4 1D Numerical Results
We first present our numerical results obtained by using EEE for optimal c and then using EFEE to find optimal mesh h which in turn is expressed by the number of interior nodal points n1. For a fixed n1 = 6, our algorithm locates an optimal c in an interval around 1 as shown in Fig. 1 with decimal digits down to 0.001 and yields approximate eigenvalues with accuracy down to the order of 10−7. Continuing the search with shrinking intervals as shown by Figs. 1, 2, 3, and 4, we see from Fig. 4 that the approximate eigenvalue can be very accurate down to an order of the machine accuracy which is about 10−16 with the optimal parameter c∗ = 1.003086658107as shown in Table 2.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 10-6
10-5 10-4 10-3 10-2 10-1 100 101 102
parameter c
EEE
←(1.003, 5.761547114957466e-006)
Figure 1: An optimal c is found around 1 by using EEE with decimal digits decreasing from 0.1 to 0.001.
1.002 1.0025 1.003 1.0035 1.004 1.0045
10-8 10-7 10-6 10-5 10-4
parameter c
EEE
←(1.003087, 2.272428822891470e-008)
Figure 2: An optimal c is found around 1 by using EEE with decimal digits decreasing from 10−4 to 10−6.
1.0031 1.0031 1.0031 1.0031 1.0031 1.0031 10-12
10-11 10-10 10-9 10-8 10-7
parameter c
EEE
←(1.003086658, 7.139178137549607e-012)
Figure 3: An optimal c is found around 1 by using EEE with decimal digits decreasing from 10−7 to 10−9.
1.0031 1.0031 1.0031 1.0031 1.0031 1.0031 1.0031 1.0031 1.0031 1.0031 1.0031 10-13
10-12 10-11 10-10
parameter
EEE
Figure 4: An optimal c is found around 1 by using EEE with decimal digits deacreasing from 10−10 to 10−12. The final shape parameter c∗ = 1.003086658107. The error of the final approximate eigenvalue is about 10−16' 0.
Table 2. n1 = 6, λ = π2
Optimal c EEE
1.000000000000000e+000 2.073648937663819e-004 1.000000000000000e+000 2.073648937663819e-004 1.003000000000000e+000 5.761547114957466e-006 1.003100000000000e+000 8.867413807678304e-007 1.003090000000000e+000 2.221198069918273e-007 1.003087000000000e+000 2.272428822891470e-008 1.003086700000000e+000 2.785013109019019e-009 1.003086660000000e+000 1.258904092082958e-010 1.003086658000000e+000 6.856737400084967e-012 1.003086658100000e+000 7.283063041541027e-014 1.003086658100000e+000 7.283063041541027e-014 1.003086658107000e+000 0
From Table 3, we observe that our methods may be able to obtain very accurate eigenvalues but the corresponding eigenfunctions may not be so accurate, e.g., the cases of n1 = 2, 4, 8, 9. In this table, the shape parameters are different. In particular, for the case of n1 = 10, we obtain an accuracy of order 10−9 for both the approximate eigenvalue and eigenfunction with the optimal c∗ = 2.099. This indicates that we need to adjust both c and h in order to obtain accurate eigenpairs.
Table 3.
n1 EEE EFEE
1 8.881784197001252e-015 4.571300945650017e-004 2 1.776356839400251e-015 2.831466309726923e-001
3 0 1.220170557396649e-004
4 5.329070518200751e-015 2.828381435379448e-001 5 2.532050381693907e-003 3.075910462934511e-005
6 0 6.946752956600577e-006
7 8.827330248983856e-005 1.374836642073496e-006 8 3.140879473129132e-005 2.828426010644248e-001 9 2.561937648692947e-007 2.828427160371329e-001 10 2.876596738587978e-010 3.895556544009082e-009
The estimators EEE and EFEE are not useful if exact eigenpairs are not available.
eigenpair with n1 = 10corresponding to the optimal parameter c∗ = 2.097999999991. For the Rayleigh quotient eλ in (2.10), the integrals are calculated by using the trapezoidal rule.
Table 4.
n1 RQEE REE
1 7.105427357601002e-015 1.471303289807391e-002 2 3.552713678800501e-015 1.277568783534476e-002 3 5.329070518200751e-015 1.359632550520781e-003 4 7.105427357601002e-015 3.988873703644078e-003 5 2.552109017846860e-003 2.410447925892204e-009 6 1.421085471520200e-014 1.081225163875287e-004 7 8.928155216381128e-005 3.709524969958490e-012 8 3.199313755430921e-005 3.739234470426920e-010 9 3.973137516766201e-007 6.422619344600665e-013 10 6.300808763626264e-010 2.336339461116005e-014
n1 EEE EFEE
1 6.307346926224255e-004 4.554340954222233e-004 2 4.329831907767812e-004 2.831492265267853e-001 3 6.419513020894385e-005 1.225448646976490e-004 4 3.671766807755716e-005 2.828382598993079e-001 5 2.532052252652406e-003 3.075911640418316e-005 6 7.341415235728732e-006 7.033255352666135e-006 7 8.830493683298357e-005 1.375531756206416e-006 8 3.143383097281571e-005 2.828426010843476e-001 9 2.561937648692947e-007 2.828427160371329e-001 10 3.032812934122831e-008 3.953132311079743e-009
3 Two Dimensional Eigenvalue Problem
In the section, we consider the equation (1.1a) in a two dimensional domain
−∆u = λu, ∀(x, y) ∈ Ω = {0 ≤ x ≤ 1, 0 ≤ y ≤ 1} (3.1)
with the boundary condition
u(0, y) = u(1, y) = 0, 0 ≤ y ≤ 1 (3.2a) u(x, 0) = u(x, 1) = 0, 0≤ x ≤ 1 (3.2b)
3.1 Meshfree Approximation
Similar to 1D case, the first step is to compute ∆ϕ(x, y)
ϕ(x, y) = (r2i + c2)−12 (3.3a)
∆ϕ(x, y) =−2(ri2+ c2)−32 + 3ri2(ri2+ c2)−52 (3.3b) From (3.3a) and (3.3b), we obtain
uh(x, y) = Again, we discretize the domain by Hypermesh and choose N = 121 from which we select h collocation points in accordance with Algorithm 1 where h << N .
For 2D problem, we thus have
−∆uh(xj, yj) = λhuh(xj, yj), j = 1, ..., n1 (3.5a) uh(0, yj) = uh(1, yj) = 0, j = 1, ..., n2 (3.5b) uh(xj, 0) = uh(xj, 1) = 0, j = 1, ..., n2 (3.5c)
or in matrix form
⎡
⎡
Since the matrices A and B are not square, we use Gaussian elimination to reduce (3.6) via (3.7) as
where αn1+1, ..., αn1+n2 are the corresponding coefficients at boundary points. Now
3.3 The Optimal Parameter c
The method of finding optimal c is the same as that of 1D. We first solve (3.8) with c = c1 in the interval [0.2], say, by set c1 = 0.1, 0.2, ..., 2.0 and locate the best c∗ = 0.7, say, according to the error estimator EEE or RQEE. We next choose [0.6, 0.8] as the next interval to search for c∗2 = 0.7, say. Then choosing [0.69, 0.71], we find the next c∗3 and so on.
For RQEE, we define
eλ = R
Ω|5uh|2dΩ R
Ωu2hdΩ (3.9)
which is calculated by using Gaussian quadrature rule.
3.4 2D Numerical Results
For a fixed n1 = 6, our algorithm locates an optimal c around 0.7 as shown in Fig. 5 within the decimal digit down to 0.001 and yields approximate solutions with accuracy down to the order of 10−5. Continuing the search with shrinking intervals as shown by Figs. 5, 6, and 7, we see from Fig. 7 that the approximate solution can be very accurate down to an order of the machine accuracy which is about 10−16 where the optimal parameter c∗ = 0.7013956922177070as given in Table 6.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 10-6
10-4 10-2 100 102 104
parameter c
EEE
←(0.7014, 2.655124940176279e-005)
Figure 5: An optimal c is found around 0.7 by using EEE with decimal digits decreasing from 10−1 to 10−4.
0.7013 0.7013 0.7014 0.7014 0.7015 0.7015
10-8 10-7 10-6 10-5 10-4 10-3
parameter c
EEE
←(0.70139569 , 1.366645818734469e-008 )
Figure 6: An optimal c is found around 0.7 by using EEE with decimal digits decreasing from 10−5 to 10−8.
0.7014 0.7014 0.7014 0.7014 0.7014 0.7014 0.7014 0.7014 0.7014 0.7014 10-13
10-12 10-11 10-10 10-9 10-8 10-7
parameter c
EEE
Figure 7: An optimal c is found around 0.7 by using EEE with decimal digits decreasing from 10−9 to 10−15. The error of the corresponding approximate eigenvalue is about 10−16≈ 0.
Table 6. n1 = 6, n2 = 40, λ = 2π2
Optimal c EEE
7.000000000000001e-001 8.648068715135793e-003 7.000000000000001e-001 8.648068715135793e-003 7.010000000000001e-001 2.442559342711093e-003 7.014000000000000e-001 2.655124940176279e-005 7.014000000000000e-001 2.655124940176279e-005 7.013960000000000e-001 1.897066400857739e-006 7.013957000000000e-001 4.795181851591224e-008 7.013956900000000e-001 1.366645818734469e-008 7.013956920000000e-001 1.351132539184619e-009 7.013956922000000e-001 1.233395607869170e-010 7.013956922200000e-001 3.437961026975245e-011 7.013956922180000e-001 4.281019982954604e-012 7.013956922177000e-001 1.172395514004165e-013 7.013956922177000e-001 1.172395514004165e-013
From Tables 7, 8, and 9, we observe that our methods may be able to obtain very accurate eigenvalues but the corresponding eigenfunctions may not be so accurate. The best result we have obtained for the eigenfunction is of 10−5 as shown in Table 9 for the case of n1 = 30, n2 = 40. The corresponding c∗ = 1.7200007999996(not shown).
Table 7. n2 = 20
n1 EEE EFEE
3 1.051603248924948e-012 3.998328795426712e-001 6 7.780442956573097e-013 9.323130199904994e-004 9 9.117577803863242e-010 3.999014151712024e-001 12 1.328714915871387e-012 4.001010057305918e-001 15 8.355982572538778e-012 4.000079339555835e-001 18 1.307398633798584e-012 1.200065872090256e-003 21 1.625116465146448e-002 4.001744898591800e-001 24 6.163848832287044e-003 6.648994853560503e-004 27 8.300204967781610e-011 3.999977978497762e-001 30 1.540847449632565e-010 3.999965811371400e-001
Table 8. n2 = 30
n1 EEE EFEE
3 3.197442310920451e-014 3.789583159383758e-001 6 3.552713678800501e-015 1.159432757862934e-001 9 1.421085471520200e-014 1.416641626055920e-001
12 0 3.777355870634687e-001
15 0 3.780150688433562e-001
18 0 9.425510785582357e-002
21 0 3.843352862618907e-001
24 0 3.763234341777008e-001
27 0 3.941652992843381e-001
30 0 6.860174189805046e-002
Table 9. n2 = 40
n1 EEE EFEE
3 5.029411731527489e-007 5.770597102014163e-004
6 0 1.479407205388798e-002
9 3.626709830051311e-004 1.372971643059315e-004 12 1.599038412791742e-008 4.001147369203052e-001 15 9.043944615427790e-008 3.999897093160651e-001 18 5.953928905455541e-009 2.503308463001619e-004 21 1.343450734282214e-003 2.959351178507541e-004 24 1.475946122763716e-002 2.491993471403844e-004 27 3.907993217922012e-008 4.000055519555411e-001 30 7.017523454777574e-006 4.201554677268210e-005
As shown in Tables 10, 11, and 12, the estimator RQEE can yield good approximate eigenvalues whereas the eigenfunctions are not so accurate in terms of REE. For example, in Table 10 with n1 = 9, and n2 = 40, the best we can get for REE (eigenfunction) is of 10−9although the error order of the eigenvalue is about 10−5. For this particular case, the shape parameter c∗ = 1.994999993000008. When compared with 1D cases, we observe that the error order of the eigenfunctions (REE) is only 10−9 for 2D problem (Table 12) whereas it is of 10−14 for 1D problem (Table 4). This enormous discrepancy is due to the boundary condition. For 1D problem, we only have two boundary points whereas there are infinitely many boundary points for 2D problem. It is well known that the major difficulty for MF methods is the problem of handling boundary conditions. Although we increase the boundary collocation points from n2 = 20 to n2 = 40, the improvement is
still very limited.
Table 10. n2 = 20
n1 RQEE REE
3 2.309263891220326e-013 1.792842369193801e-004 6 1.989519660128281e-013 5.903594518448732e-005 9 1.847819675049323e-009 2.191935284842237e-008 12 4.986944190932263e-011 5.258122603373290e-006 15 1.137223648584040e-011 4.941026922158529e-006 18 2.803091092573595e-012 1.585124853257117e-004 21 1.735706603356135e-002 9.371643390468916e-008 24 6.394884621840902e-014 1.107637934417366e-001 27 2.470095239459624e-010 2.596437210483295e-006 30 2.358504502808501e-010 8.156003764574335e-007
n1 EEE EFEE
3 8.971711875496169e-003 3.998025956959469e-001 6 8.852748011481282e-003 7.339660698468531e-004 9 6.416475886023676e-003 2.091870263639523e-004 12 4.142392995184707e-003 3.222483635615947e-004 15 4.445056143005388e-004 4.000077632665167e-001 18 5.812704947224745e-003 4.001939813621466e-001 21 1.625204338048292e-002 2.718101856637595e-004 24 2.086568581023158e-001 4.003952956721543e-001 27 5.262281742091091e-004 3.999997890426761e-001 30 3.600247718296146e-004 2.338443874675648e-004
Table 11. n2 = 30
n1 RQEE REE
3 2.220798075214248e-006 1.936710019497395e-003 6 7.897240195120503e-009 1.826116256389247e-004 9 2.691227469659232e-001 1.074894876540476e-009 12 7.525768985772174e-008 2.663044246103444e-005 15 7.887681618967690e-010 4.964559762588703e-005 18 1.168155172592833e-008 7.949323108390297e-006 21 1.246571734725421e-008 2.574665389323098e-005 24 8.327954414966143e-005 1.011620118236780e-006 27 4.808444820803004e-005 1.879831275846143e-007 30 1.205087462707866e-006 2.265002867504518e-006
n1 EEE EFEE
3 7.409256117177664e+000 3.448342121922024e-001 6 7.400546688656135e+000 3.463463760467250e-001 9 1.573404747963234e+001 3.065954122174966e-001 12 7.408521960784686e+000 3.445170812234079e-001 15 7.403297676512377e+000 3.445984488879295e-001 18 7.404996215798031e+000 1.905719018820360e-001 21 7.397640365182175e+000 1.915723607534694e-001 24 7.407085348199436e+000 3.445314041266190e-001 27 7.402681914888055e+000 1.906919557387647e-001 30 7.402996432161485e+000 3.444687781680308e-001
Table 12. n2 = 40
n1 RQEE REE
3 8.479816884232605e-007 1.409511869824942e-004 6 6.730361697293574e-006 1.605625503236164e-005 9 9.115163980411012e-005 3.606837700188802e-009 12 4.386487972851683e-008 5.036766504062181e-006 15 4.328121505636773e-007 1.632792488775396e-007 18 1.244221436991211e-008 3.594343389792533e-007 21 6.118882560105732e-003 1.133347651194547e-008 24 1.488587031417410e-012 1.254943471350926e-002 27 5.754479559527681e-009 8.769810491254240e-007 30 1.230076290781312e-005 1.062186809622102e-008
n1 EEE EFEE
3 1.355083068722962e-002 3.997379057528611e-001 6 1.454215841452466e-002 8.913387879772861e-004 9 9.376626077521877e-003 3.998634633960277e-001 12 6.171429215260815e-003 3.728133554017910e-004 15 2.707928547984295e-003 3.999887578175357e-001 18 1.004959669735683e-003 4.000354201684913e-001 21 1.343450734282214e-003 2.959351178507541e-004 24 1.987769446649530e-001 1.249300359026555e-002 27 4.272305695813827e-004 9.443932287149515e-005 30 1.098144200259554e-003 3.999927113531166e-001
4 Three Dimensional Eigenvalue Problem
The 3D problem is
−∆u = λu, ∀(x, y, z) ∈ Ω = {0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} (4.1) with the boundary condition
u(0, y, z) = u(1, y, z) = 0, 0≤ y ≤ 1 ,0 ≤ z ≤ 1 (4.2a) u(x, 0, z) = u(x, 1, z) = 0, 0≤ x ≤ 1 ,0 ≤ z ≤ 1 (4.2b) u(x, y, 0) = u(x, y, 1) = 0, 0≤ x ≤ 1 ,0 ≤ y ≤ 1 (4.2c)
4.1 Meshfree Approximation
For 3D case, we have
ϕ(x, y, z) = (ri2+ c2)−12 (4.3a) We choose N = 1331 for small h and thus have
−∆uh(xj, yj, zj) = λhuh(xj, yj, zj), j = 1, ..., n1 (4.5a)
4.2 Gaussian Elimination
Since the matrices A and B are not square, we use Gaussian elimination to reduce (4.6) via (4.7) as
square matrix system
4.3 The Optimal Parameter c
Again, we use the Algorithm 2 to find the optimal shape parameter c. We first solve (4.8) with c = c1 in the interval [0, 2], say, by setting c1 = 0.1, ..., 2.0 and c∗1 = 1.8, say, according to the error estimator EEE or RQEE. We next choose [1.7, 1.9] as the nest search for c∗2 = 1.82, say. Then choosing [1.81, 1.83], we find the next c∗3 and so on. For RQEE, we need to evaluate
eλ = R
Ω|5uh|2dΩ R
Ωu2hdΩ (4.9)
by using again the Gaussian quadrature rule.
4.4 3D Numerical Results
For a fixed n1 = 27, our algorithm locates an optimal c in an interval around 1.8. Contin-uing the search with shrinking intervals, we obtain approximate solution with accuracy down to the order about 10−9 with the optimal parameter c∗ = 1.818458802000001 as
shown in Table 12.
Table 12. n1 = 27, n2 = 120, λ = 3π2
Optimal c EEE
1.800000000000000e+000 1.405729241561460e-003 1.820000000000000e+000 1.158633573368206e-004 1.818000000000000e+000 3.430525096703718e-005 1.818500000000000e+000 3.259977862768437e-006 1.818460000000000e+000 1.587118099166673e-007 1.818458000000000e+000 1.734633059413682e-008 1.818458800000000e+000 7.796312928576299e-009 1.818458800000000e+000 7.796312928576299e-009 1.818458802000001e+000 1.292786322437678e-009
From the experience of 1D computations, we can obtain accurate eigenpairs by seeking optimal parameters c, and h. Form 2D, we know the accuracy is greatly affected by the number of boundary collocation points. With n2 = 120, Table 13 shows that the best eigenvalue is O(10−13) for n1 = 18 and the best eigenfunction is O(10−5) for n1 = 27.
Increasing the boundary points to n2 = 602, Table 14 shows that the best eigenvalue and eigenfunction we can get is of O(10−9) and O(10−5), respectively, with n1 = 27 and c∗ = 1.7502975. Note that the larger number of collocation points n1 + n2 does not necessarily imply more accurate eigenpairs. This is another well known fact for RBF-MF methods that, after a certain number of degrees of freedom, the algebraic system becomes more ill-conditioned with more basis functions. It is even worse when we use the error estimators RQEE and REE as shown by Tables 15 and 16. Moreover, the computational complexity of Gaussian elimination is O(n32) which means that the more boundary collocations the slower solution process. Obviously, there are much more works
for 3D eigenvalue problems in the future.
Table 13. n2 = 120, λ = 3π2
n1 EEE EFEE
3 7.078749775999640e-001 2.112564972010275e-003 6 1.038690224188201e-001 1.796180269205369e-001 9 6.710602134360894e+000 7.212026441604076e-003 12 7.994795201959448e-003 7.560564224688606e-004 15 5.723009855926254e+000 9.231128037832083e-003 18 8.810729923425242e-013 4.228559596524420e-003 21 2.443524701976330e-002 5.749648150754800e-003 24 5.466098684792087e-010 1.789558035031090e-001 27 1.522479919913167e-009 8.311505134660702e-004 30 3.931699410486544e-009 1.883960546458598e-003
Table 14. n2 = 602, λ = 3π2
n1 EEE EFEE
3 4.937493984768544e-001 1.793288990738743e-001 6 1.814615158847097e-001 1.798650517574291e-001 9 3.777679267957776e+000 1.801034703708724e-001 12 6.830999976340380e-002 6.373780342875390e-004 15 3.706526931535553e+000 1.361058491322728e-002 18 9.393635240648735e-003 2.182444476564530e-004 21 4.620000494526266e-005 1.790313799495266e-001 24 2.084964734194728e-007 1.787773649474201e-001 27 7.626248077485798e-009 7.690443366782304e-005 30 3.469220364305414e-003 1.789871432924734e-001
Table 15. n2 = 120
n1 RQEE REE
3 6.925333282582216e-001 1.964000639475076e-005 6 1.131639539948388e-001 1.800799410329172e-005 9 6.795859361403430e+000 1.452564108133618e-005 12 3.961698524790336e-009 7.069377195609461e-006 15 6.130667563553249e+000 1.409919919389726e-007 18 1.112780978473893e-010 4.525486438312680e-004 21 5.419735771283740e-010 5.086660311853171e-005 24 1.313111752665463e-008 3.624089272296941e-006 27 5.017035675791703e-010 1.689665119787359e-005 30 8.471552348510159e-010 1.275468796517158e-005
n1 EEE EFEE
3 6.963492430296974e-001 2.103247290360565e-003 6 1.150286552003053e-001 1.795961268627190e-001 9 7.044392768485214e+000 8.025845797520706e-003 12 1.605072818471598e-002 1.792075170408243e-001 15 6.271288804842857e+000 6.171404329094042e-003 18 3.744951161475996e-002 3.777790502618356e-003 21 6.771504815773710e-003 1.789288319892565e-001 24 2.043543298136896e-002 1.789380351159896e-001 27 1.293815794707953e-002 8.037844809589339e-004 30 3.194354950068146e-003 1.792978756467908e-001
Table 16. n2 = 602
n1 RQEE REE
3 4.866465193228642e-001 3.547177434240611e-006 6 1.910831887274860e-001 1.855101750064742e-005 9 2.871907649259498e+000 1.286061054420139e-004 12 1.157798543045630e-002 1.294466023931101e-006 15 3.308887760791865e+000 1.879758359507373e-004 18 1.033839680530946e-012 1.423746203439977e+000 21 1.546527936824305e-006 1.174819358552902e-006 24 4.390435996981523e-007 1.927691861352076e-006 27 4.909523017460060e-006 1.114580822037810e-006 30 7.757033327848717e-005 4.114250624377626e-008
n1 EEE EFEE
3 4.937493984768544e-001 1.793288990738743e-001 6 1.814956128800596e-001 1.797716277277273e-001 9 4.028345939579598e+000 1.791173038787475e-002 12 2.871402194960737e-002 1.790769823604051e-001 15 4.215652814071181e+000 1.600050523945558e-002 18 6.082291194123322e-001 6.272140019894984e-003 21 1.314005422386799e-003 3.423753602635471e-004 24 2.438126020685161e-004 1.051413359443865e-004 27 2.121500925603925e-003 6.709598235409356e-005 30 1.575629506520571e-002 3.277554002275585e-004
5 Conclusion
In this work, we propose a radial basis functions based mesh free method (RBF-MF) for elliptic eigenvalue problems in 1, 2, and 3D. The discrete eigenvalue matrix system is obtained by means of Gaussian elimination of matrix coefficients associated with the boundary collocation points. Four error estimators (EEE, EFEE, REE, RQEE) are given to determine the optimal shape parameter and the number of collocation points under certain error tolerance. The main idea for obtaining accurate eigenpairs is to devise error estimators and algorithms to search the optimal shape parameter with very many decimal digits. We summarize our findings as follows:
1. It is shown for 1D and 2D problems that the error order of the approximate eigenvalues λh can be as low as O(10−16)' 0 which is the machine limit of Matlab with the optimal shape parameter c up to 16 decimal digits by using EEE. The best case for
the 3D problem is O(10−13).
2. Using EEE to find optimal c and using EFEE to find optimal mesh size h, the best we can get for the eigenfunction uh is O(10−9) and for the eigenpair (λh, uh) is (O(10−10), O(10−9)) for 1D problem.
3. Using RQEE to find optimal c and using REE to find optimal mesh size h, the best we can get for λh, uh, and (λh, uh)are respectively O(10−15), O(10−14), and (O(10−10), O(10−14)) for 1D problem as shown in the following table where the similar data for 2D and 3D problems are also included.
EEE λh EFEE uh RQEE λh REE uh
1D λh 0 O(10−15)
1D uh O(10−9) O(10−14)
1D (λh, uh) O(10−10) O(10−9) O(10−10) O(10−14)
2D λh 0 O(10−14)
2D uh O(10−5) O(10−9)
2D (λh, uh) O(10−6) O(10−5) O(10−5) O(10−9) 3D λh O(10−13) O(10−12)
3D uh O(10−5) O(10−8)
3D (λh, uh) O(10−9) O(10−5) O(10−5) O(10−8)
4. Accuracy is greatly affected by the number of boundary collocation points for 2D and 3D problems.
5. Accuracy associated with RQEE is influenced by the Gaussian quadrature rule.
6. After a certain number of degrees of freedom, the algebraic system becomes more ill-conditioned when the number of basis functions is increasing.
7. The computational complexity of Gaussian elimination is O(n32)which means that the more boundary collocations the slower solution process.
8. More works are needed for 3D eigenvalue problems with irregular domains in the future studies.
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