Improved Multiquadric Method for Elliptic Partial Differential ... - CORE

An International Journal

computers & mathematics PERGAMON

Computers

and Mathematics

with Applications

43 (2002) 439455 www.elsevier.com/locate/camwa

Improved Multiquadric Method for Elliptic Partial Differential Equations via PDE Collocation on the Boundary A. I. FEDOSEYEV Center for Microgravity and Materials Research University of Alabama in Huntsville Huntsville, AL 35899, U.S.A. [email protected] M. J. FRIEDMAN Department of Mathematical Sciences University of Alabama in Huntsville Huntsville, AL 35899, U.S.A. friedmanQnath.uah.edu E. J. KANSA Embry-Riddle Aeronautical University East Bay Campus, Oakland, CA 94621, U.S.A. kansaiQllnl.gov Abstract-The multiquadric radial basis function (MQ) method is a recent meshless collocation method with global basis functions. It was introduced for discretizing partial differential equations (PDEs) by Kansa in the early 1990s. The MQ method was originally used for interpolation of scattered data, and it was shown to have exponential convergence for interpolation problems. In [l], we have extended the KansaMQ method to numerical solution and detection of bifurcations in 1D and 2D parameterized nonlinear elliptic PDEs. We have found there that the modest size nonlinear systems resulting from the MQ discretization can be efficiently continued by a standard continuation software, such as AUTO. We have observed high accuracy with a small number of unknowns, as compared with most known results from the literature. In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the boundary (MQ PDECB): we add an additional set of nodes (which can lie inside or outside of the domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. Numerical results are given that show a considerable improvement in accuracy of the MQ PDECB method over the Kansa-MQ method, with both methods having exponential convergence with essentially the same rates. @ 2002 Elsevier Science Ltd. All rights reserved.

Keywords--Radial basis functions, furcations, Nonlinear elliptic PDEs.

Multiquadric

method, Numerical solution, Continuation,

Bi-

This work was supported in part by National Aeronautics and Space Administration through Grant NAG&1229. The first author is grateful to V. N. Belykh for fruitful discussions, that led us to the idea of the PDECB. 0898-1221/02/$ - see front matter @ 2002 Elsevier Science Ltd. All rights reserved. PII: SO898-1221(01)00297-8

Typeset by -&$-‘IBX

440

A. I. FEDOSEYEV

et al.

1. INTRODUCTION The multiquadric collocation

radial

method,

basis function

with global basis functions,

in 1970 [2,3] for interpolation convergence

(MQ RBF or, simply,

for function

of scattered

approximation.

for discretizing

data

in [7,8] in the early 1990s. Since then it was successfully based directly

on the interpolation

of the MQ method mention

a recent

yielding

banded

to PDEs

paper

error estimates,

method

matrices

of arbitrary

meshless

It was originally

was introduced

applied

appeared

leads to finite-dimensional

is a recent

proposed

[4-61 to have an exponential for solving

while some convergence

[17], where MQ basis functions

collocation

The Kansa-MQ

therein,

PDEs.

and was shown

The MQ method

3D PDEs, see, e.g., [g--14] and references

MQ) method

a number

of 2D and

[15,16]. Application

with full matrices.

with compact

PDEs

results for solving PDEs,

only recently

problems

for solving

support

We also

were constructed,

band width.

was shown to give high accuracy

with a relatively

small

number

of

unknowns (tens or hundreds for 2D problems). The corresponding linear systems can be efficiently solved by direct methods. In [l], we have extended the Kansa-MQ method to numerical solution of parameterized nonlinear elliptic PDEs. We presented there results of our numerical experiments with continuation of solutions to and detection of bifurcations in 1D and 2D nonlinear elliptic PDEs. We found that the modest size nonlinear systems resulting from the MQ discretization can be efficiently Our observations one to two orders)

continued

by a standard

continuation

software,

such as AUTO

[18].

have shown that the residual error is typically largest near the boundary compared to the residual error in the domain far from the boundary.

(by

In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the we add an additional set of nodes (which can lie inside or outside of the boundary (PDECB): domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. The motivation for this modification of the Kansa-MQ (1) the residual

method

comes from our observations

is typically

the largest

that

near the boundary

(by one to two orders

in the domain far away from the boundary), and (2) the residual is dramatically reduced when we use the PDE collocation

larger than

on the boundary.

The MQ PDECB method leads not only to a higher accuracy, but, for nonlinear problems, also to a higher eficiency due to the reduction of the number of unknowns in the continuation process by using a preprocessing. We apply

our MQ PDECB

PDEs and present

results

method

to several

of our numerical

model

experiments.

improvement in convergence of the MQ PDECB method methods having exponential convergence with essentially is the first demonstration of the exponential convergence

1D and 2D linear

and nonlinear

elliptic

These results demonstrate considerable over the Kansa-MQ method, with both the same rates. To our knowledge, this for the MQ method applied to PDEs.

A related idea was successfully used for high Re number fluid flows in the cases of the RNS model [19,20] and the Alexeev hydrodynamics equations [21] (in the framework of the finite element method) for the solution of 3D thermo-vibrational flows [22]. A class of global numerical methods for 1D and 2D problems, the numerical algorithms without saturation, was proposed in the early 1980s in [23]. This class includes a highly accurate discretization method for PDEs based on Chebyshev polynomials. This method was further developed in [24,25], where it was found to be more accurate and better conditioned than the spectral

method.

In Section 2, we formulate the Kansa-MQ and the MQ PDECB methods for a linear elliptic PDE. In Section 3, we describe in detail the Kansa-MQ and the MQ PDECB methods for continuation of solutions to parameterized nonlinear elliptic PDEs. For clarity of presentation, Section 3 is written independently of Section 2. In Section 4, numerical examples are given that illustrate the accuracy of our method. In Section 5, we summarize our results.

441

Improved Multiquadric Method

2.

Weconsidera well-posed

A LINEAR

ELLIPTIC

elliptic boundary value problem: for given g(z), f(x) find U(Z) from LG)

C !R*, (1)

= g(z),

where R is a bounded domain with the boundary operator, and B is a boundary operator.

Introduce

in R

= f(z),

%&,

2.1.The Kansa-MQ

PDE

HI, L is a linear elliptic partial differential

Method

a set Oh of nodes (Figure

1)

and the MQ basis functions

gjw = sj (Cj,x) = & - Zjll$ + c;, j = l,...,

i’/ + Nb,

$v+jvb+l(z) = 1,

(3)

I.IIRd is the Euclidean norm in R*, cj 2 0 are called shape paramewhere x, LC~E B* and ((z - CT ters [8]. We look for the approximate solution uh to (1) in the form N+Nb+l

W(X) =

C

(4

Ujgj(X).

jd

Substituting problem

L

’

Us

into (1) and using collocation at the nodes @h, we obtain the finite-dimensional

=N~lU_fLgj(X~)xf(Xi),

rElujLlj(Xi))

i=l,...,N,

rrlajg.itxi))

i=N+l,...,N+Nb,

=N~'aj3~j(xi)=g(xi)7

N-tNb

c = aj

0.

j=l

Nb ’

u/

0 N+N,+l I h,

N,

2 ... N m m w u/

N+l A -

L-h-4 1

2 .,.

N m

0

N+l

N+2N, I

1 (a) 1D nodes.

Figure 1. Nodes for the Kansa-MQ and MQ PDECB methods: (a) 1D Kansa-MQ nodes (top) and PDECB (bottom), node numbering is shown; (b) 2D Kansa-MQ nodes (left) and PDECB (right): o-nodes for PDE collocation; l-BC collocation; a--PDE and BC collocation; -i-nodes added for PDECB; hl is a distance to the boundary (may be negative, if nodes are inside); h is a mean distance between nodes.

(5)

442

A. I. FEDOSEYEV et al. +

+

+

+

+

0

0

0

f

0

0

0

f

+

+

+

+

+

+

(b) 2D nodes. Figure 1. (cont.)

Introducing &N+NJ,

a = (or,. . . , ary+~~+l)~,

the notation

and 13= (f(zr),

.

qT E RN+Nb+l,

&?N+tc,+l

w=

Bg = BSN+Nt,(ZN+&)

(6)

(xN+l)

&N+&+lh’+N,)

4 [4

1’

0 we can rewrite

the system

(5) in the matrix

form as Wa=b,

whose solution

(7)

is LY= W-lb.

2.2. The Introduce

MQ

PDECB

a set 0;

(8)

Method

of nodes (see Figure

1)

which can be inside G or outside where the nodes {zi}~?,$$~+r, ary dR, and the MQ basis functions

s.?(x)= Sj(Cj,x>= Jllx- X&d + c$ We look for the approximate

solution

j

=l,*..,N+Nb,

a, are adjacent

gN+2N,,+l(X)

to the bound-

= 1.

(10)

uh to (1) in the form

(11) Substituting problem

L

Us

into (1) and using collocation

r+gYf’,,l.d)

=N+~+'.jLgj(x~~=i(..,,

N-!-2NbS1 B

at the nodes Ok, we obtain

c

=

ajgj(xi)

N+sNb+l c ajBgj(xi)

N+2Nb

c = aj

j=l

0.

the finite-dimensional

i=l,...,N+N&

= g(xi),

i = N + -,...,

N +Nb,

(12)

Improved Multiquadric

the notation a = (a~, . . . ,uN+xK,+I)~,

Introducing

g(xN+2Nb);O)T

E

Method

b = (f(zl), .-.

443

. . I ,f(znr+~,),g(~clv+~,+1),

RN+2Nb+1,

1 J

LgN+2Nb+l(zd

>

I

Lh’+2h’b+1

&l(zN+l)

B,= .7

*‘*

BgN+2N,(zN+l)

i

i

Bgl (xN+& >

’ *.

1

...

we can rewrite

the system

(xh’+h’b)

1

&V+2N,+l(zN+l)

BgN+2Nb(xN+Nt.) 1

(12) in the matrix

’

&N+2Nb+l(xN+Nt,) 0

(13)

12 I,

w=

9

form as Wa=b.

3. CONTINUATION Consider elliptic

...,

a boundary

FOR

value problem

(14

NONLINEAR

for a second-order

ELLIPTIC

system

PDES

of n parameterized

nonlinear

PDEs F (u(x), X) = D(x)Au

B+)l,,

- f(Vu,

U,Z, X) = 0,

in R c II?+, x E I[$, u(.) E I[$“, (15)

= 0,

where R is a bounded domain, D(X) is a positive diagonal n x n matrix, f is smooth, boundary operator which we assume, for simplicity, to be linear. For the bifurcation the process of continuation, we also need to consider DrF(u,A) of F about the solution u of (15) DiF(u,

@+)

the eigenvalue

problem

and B is a analysis in

for the linearization

in 52,

= /4x),

(16)

Bw(z)(~~ = 0. 3.1. The Kansa-MQ Method To formulate

the approximate

problem,

we first introduce

the set Qh of nodes

Oh = { (Xi),N=i c S-4 {xi>:$,

c asl}

(17)

and the MQ basis functions, Sj(X) = &j,

x) = JIIW,

j =

l,.**,N+Nbr

The same points xi, i = 1,. . . , N + Nb will be used as the collocation MQ finite-dimensional subspace

sh:=

x= (

Problems

N+h’b+l c ajgj(.): j=l

h’+Nb c uj=O,

&F(wz,

1.

(18)

We next define an

Bx(xi)=O,i=N+l,...,N+Nb

are approximated

F (~h(zi)r =

points.

=

(1%

j=l

(15) and (16), respectively,

Luh(xi)

gN+Nb+l(x)

~w(G)

4 = 0, = pub(%),

by the collocation

equations

UhE,.!&,

i=l,...,

N,

(20)

VhESh,

‘i=l,...,

N.

(21)

A. I. FEDOSEYEV et al.

444

Substituting N+Nb+l %(~I

=

c

aj9jcG

(274

bjSjc4,

(23)

j=l N+Nb+l %b)

=

c 3=1

into (20) and (21), respectively, and using the definition (19) of Sh, we obtain the following finite-dimensional problems:

i=l,...,N, N+Nb+l

i=N+l,...,N+Nb,

(24)

NfNb

c =

0,

aj

j=l

N+Nh+l

N+Nb+l

c

W9j(G)

c

bjB9j (Xi) = 0,

= I-L

j=l

c

i=l,...,N,

bjSj(Xi),

j=1

i=N+l,...,N+Nb,

(25)

j=l

N+Nb

bj =O.

c j=l

Introducing the notation a = (al,. . . , aN+Nh+l)T, b = (bl,. . . , bN+Nb+l)T E @?x(N+Nh+l), [

Bgl(xN+l)

‘.’

B9N+Nb

(xN+l)

:

: . . . Bg,N- tNbbN+Nb)

1

...

Bgl(x~+~,,)

L9N+Nt,+l

B9N+Nb+l(ZN+l)

1

BSN+Nb+lbN+Nf>)

1

0

(26)

(21)

I7

1

we can rewrite problems (24) and (25) in the matrix form as G(a, A) = 0, L,b = prb, Implementation

B,a = 0,

(27)

B,b = 0.

(28)

1

Let

a1 = (al,...,aN)T EWnxN,

a2 = (aN+l,...

, aN+Nb+l)TE

. ’ ’

BnxcNb+‘);

LgN+Nb+l(ZN)

Improved Multiquadric Method 91(x1) : Sl(ZN)

*..

9Nh)

;

;

...

9N(~N)

1 [

9N+lh)

,

r2=

;

this into (27), we rewrite

...

SN+N*+l(Q)

i

9N+lkN)

!

Substituting

*. *

..*

SN+Nb+l(xN)

I

445

i

it as

(29)

6 (a’, A) 3 G (a’, u2, A) = 0, where a2 solves B2a2 = -@a1 9 9 Similarly,

we rewrite

(30)

.

(28) as L;b’ + Lib2 = p (I”b’

+ r2b2) ,

B1bl 9 + B2b2 9 = 0. b2, as

or, eliminating

Lib’ - L; (Bg”)-‘B;b’

= p @‘lb’

+ r2 (Bg”)-’ Bib’)

We are interested in the continuation of solutions to (29). Therefore, treat X as unknown, and add an algebraic constraint

.

in addition

(31) to ul, we also

G, (a’, A) = 0, which defines

a parametrization

ALGORITHM 1. (Continuation

of the solution algorithm

to a1 E WnxN and X E IR, a complete

curve.

for the system

Newton

(32)

iteration

(29) ,( 32) .) Given current

approximations

consists

steps.

of the following

(0) Compute the matrices Bi, B2 I? 7 IT2’ (1) Solve the system (30) to findgi2. (2) Use the expressions (29),(32) to compute

the residuals +(a’, compute the matrices DrQ = DrG(a’,X), DzQ f D&(u’,X), DzG, E D2G(a1, A) by differencing. (3) Solve the system D&&z1 + D&6X = -G (a’, A) , DIG,Gal + DzG,GX = -G,

A), -G,(al, A), and then DrG, E DrG,(ar,X), and

(33)

(a’, A) ,

where we omitted iteration indices for ba’ and 6X in (33). u1 ---f a1 + Sal and X -+ X + 6A. Solve the generalized eigenvalue problem (5) (4) Update

DlOb’ = /.A (I” + r2 (B,“)-’ to detect

bifurcations.

Note that

B;)

b’

DIQb’ = Ljb’ - Lz (Bi)-l

(34)

Bg’b’; see (31).

A. I. FEDOSEYEV et al.

446

Implementation

2

Let U = (Vi,. . . , VN)~ be the vector of nodal values of the solution uh (22) of the collocation problem (20), and let {&}g,

be the Lagrange basis in sh

{$bj ‘? sh : $j(%i) = C&j,i,j = 1,. . ,N} .

(35)

Then ?_&can be written as uh(x)

= 2

(36)

uj$j(z).

j=l

Combining this with the definitions (22) of uh and (26) of B, and l?, we have l?a = U, (37)

B,a = 0, that defines the one-to-one correspondence between U Problems (20) and (21), respectively, are written as f?(U, A) = @a, A) where a solves (37), and mY(V,

X)V =

pv,

v

E

lPXN.

(39)

As before, to define a parametrization of the solution curve, we add an algebraic constraint G&J, A) = 0.

(40)

ALGORITHM2. (Continuation algorithm for the system (38),(40).)

Given current approximations

to u E TWnxNand X E R, a complete Newton iteration consists of the following steps.

(0) Compute the matrices B,, I?. (1) Solve the system (37) to find a. (2) Use the expressions (38),(40) to

compute the residuals -G(U,X),

compute the matrices 016 E DI~(U,X), D& = D&‘=(U, X) by differencing.

(3) Solve

D2G = D&(U,X),

-G,(U, A), and then

DIQc E DlS,(U,X),

and

the system DILXJ + D&6X = -6(U, A), DIG&J

+ D2G,GX = -&(U,

A),

(41)

where we omitted iteration indices in (41) for 6U and 6X. (4 Update U 4 U + 6U and X -+ X + SX. (5) Solve the eigenvalue problem (39) (to detect bifurcations). REMARK 1. For our numerical experiments, we implemented in AUTO [18] Algorithm 2 for the Kansa-MQ method and Algorithm 2a, below, for the MQ PDECB method. The principal reason for choosing Algorithm 2 rather than Algorithm 1 is that the eigenvalue problem (39) (and (56)) is a standard eigenvalue problem whose solution is supported by AUTO. On the other hand, the eigenvalue problem (34) is a complicated generalized eigenvalue problem whose solution is not supported by AUTO. 3.2. The MQ PDECB

Method

To formulate the approximate problem, we first introduce the set oh of nodes (42)

Improved Multiquadric Method where the nodes

{si)~~$$~+r,

447

which can be inside Sz or outside

n, are adjacent

to the bound-

ary dR, and the MQ basis functions, $Jj(Z> = $?j(Cj,~) = 4-T We remark

here that

j = l,..*

only the points

as the collocation

points.

as the collocation

points

In particular,

for both the PDE and the boundary

X=

=

=

1.

(43)

condition.

set (which is not a subspace,

in general)

N+2Nb+r I’.‘+ZNb C f.Zjgj(*): C aj = 0, j=l

{ Bx(zi)

gN+2&+1(2)

xi, i = 1,. . . , N + Nb, that lie in 52 and on X2 will be used the points xi, i = N + 1,. . . , N + Nb, on Xl will be used

We next define an MQ finite-dimensional

SA :=

Nb,

,N +

j=l

(44)

0, F (x(x&

i’t) = 0, i = N + 1,. . . , N + Nb

. I

Problems

(15) and (16), respectively,

are approximated

F (~&i)r Lw&i)

= DrF(~hr

by the collocation

8 = 0,

X)V&)

= ~r.+i),

equations

u&S;,

i=l,...,

N,

(45)

,uh E S;,

i=

N.

(46)

l,...,

Substituting N+Zh’b+l U/L(~) =

C

aj!lj(X),

(47)

bjSj(X),

(48)

j=l N+2h’b+l W(X)

=

C j=l

into (45) and (46), respectively, dimensional

and using definition

(44) of Si, we obtain

the following

finite-

problems: N+2Nb+l

(G(a,X))i

E F

C i

ajgj(G),X

= 0,

i = l,...,N,

= 0,

i = N + 1,. . . , N + Nb,

j=l N+2Nb+l

(G(a,X))i

z F

C

ajilj(Xi),

A

j=l

(49)

N+%%+l

C

ajBgj(Xi) =

i = N + 1,. . . , N + Nb,

0,

j=l

N+2Nb c j=l

aj = 0,

N+2&+1 C j=l

N+2Nb+l *jLgj(Xi)

=P

=

N+2&+1

c

C j=l

0,

bjBgj(Xi)

j=l

N+2Nb

c~

bj ~0.

j=l

bjgj(Zi),

i=l,...,N+Nb,

i=N+l

, . . . , N + Nb, (50)

A. I. FEDOSEYEV et al.

448

Introducing the notation a = (al,. . . ,cIN+~N,,+~)~, b = (bl, . . . , bN+zN,+l)T E IRnx(N+2Nb+l), %N+2Nt, (zN+l)

; ;

~g1(x1) L,=

r=

;

BSN+2Nb+l

(XN+N,,)

’

LgN+2Nb+1(x1)

(51) LgN+2Nb+l

cxN+Nt,)

gN+2Nb+l(x1)

i

[ 91 (XN+N~)

we

’ ’ ’

...

(XN+N~)

(xN+l)

1

.*.

[ k?l (ZN+Nb) $Il(Xl)

BSN-t2Nb

BgN+2&+1

. . *

gN+2Nt,+l

i (ZN+Nb)

1.

can rewrite problems (49) and (50) in matrix form as (G(o, X))i = 0,

i=l,...,N,

(G(o, Wi = 0,

i = N + 1,. . , N + A$,

(52)

B,a = 0, L,b = pl?b, (53)

B,b = 0. Implementation

2a

Let u = (z&(q), . . ) uh(x~))~ be the vector of nodal values of the approximate solution ‘zL~. Then by the definitions (47) of Uh and (51) of B, and l?, we have (G(o) A)), = 0,

i=N+l,...,N+N/_),

Pa = U,

(54)

B,a = 0, that defines the one-to-one correspondence between U E lRnxN and a E lRnx(N+2Nb+1). Problems (45) and (46), respectively, are written as i=l,...,N,

(55)

v E iRnxN.

(56)

(WJ, X))i = (G(o, Wi = 0, where a solves (54), and DlS(U,

X)V = Pv!

As before, to define a parametrization of the solution curve, we add an algebraic constraint G&J, X) = 0. ALGORITHM

2a. (Continuation

tions to U E RnxN

algorithm for the system (55),(57).)

(57) Given current approxima-

and X E R, a complete Newton iteration consists of the following steps.

(0) Compute the matrices B,, I’. (1) Solve system (54) to find a. (2) Use expressions (55),(57) to compute

the residuals -G(U, X), -&7,(U, A), and then compute the matrices 019 = Dl&i!(U,X), D& z D&(U,X), DIG, E D1&(U, A), and D& s D2Gc( U, X) by differencing. (3) Solve the system D166iY f D2CTSX= 4(U, A), (58) DIG&J + D2G,6X = -&(U, A), where we omitted iteration indices in (58) for HJ and 6X. (4) Update U -+ U + 6U and X --+ X + 6X. (5) Solve eigenvalue problem (56) (to detect bifurcations).

Improved Multiquadric Method

449

4. NUMERICAL EXPERIMENTS FOR 1D AND 2D ELLIPTIC PDES We present

examples

to nonlinear method

of solution

(see equation method

of the detection of equation

problems,

of the limit point

parametrization

Each problem method

we perform

and by Algorithm

f(u,x)

equation.

(38)) and the MQ PDECB

In the case of nonlinear Kansa-MQ

of linear 1D and 2D elliptic PDEs and continuation

1D and 2D Gelfand-Bratu

(see equation

continuation

method.

(or fold) by the two methods. limit

point

s, makes a turn at (ua, X0). This is expressed

h = l/(K

and for a 2D problem

on (0,l)

number of nodes along each axis. To improve the accuracy, we employ two simple

2 for the

the accuracy

curve in (u(s), X(s)),

formally

h for the average

by Algorithm

We compare

We recall that a solution

if the solution

and fx(uo,Xo) 4 NL(~o,~o)). We will use throughout, the notation

of solutions

by the Kansa-MQ

(55)).

of solutions

2a for the MQ PDECB

= 0 is a (simple)

- 1) for a 1D problem

is discretized

distance

as dimN(f,(us, between

on (0,l)

(uo, Xa) for some Xc)) = 1

the nodes.

x (0, l), where

Then K is the

adaptation strategies for the shape parameters

C = {ci,. . , CN+N,,} for the Kansa-MQ method (see equation (18)) and C’ = {cl,. . . , CN+~N~} for the MQ PDECB method (see equation (43)); for the nodes Oh for the Kansa-MQ method, see equation (17), and 0; for the MQ PDECB method, see equation (42). To be specific, assume method. Let T(Z, y, C, Oh) be that R = (0,l) x (0,l) and consider the case of the Kansa-MQ the residual. Our strategies are all based on the nonlinear least squares method which minimizes By the quasi-uniform distribution of nodes, we the Lz-norm cp(C,Oh) - llrlls of th e residual. will mean the distribution of nodes, where the nodes adjacent to the boundary dfi are placed at the distance h = 6ho, 0 < 6 5 1, from as2, while the remaining nodes are distributed uniformly with the distance STRATEGY

ho between

1. Uniform

of nodes @h; cl = . . . = CN+N* = c; min, cp(C, Oh).

distribution

STRATEGY 2. Quasi-uniform In all examples

them.

distribution

of nodes Oh; cl = . . . = CN+N~ = c; min,,a cp(C, Oh).

below, we use the adaptation

Strategy

2.

= 0,

in R = (0, l),

EXAMPLE 1. 1D MODEL LINEAR PROBLEM. u,,

+ (27r)2 sin(27rz)

(59)

u(0) = u(1) = 0. The analytical

solution

is u exact - sin(27rz).

Numerical

results

are presented

in Figure

2a.

EXAMPLE 2. 2D MODEL LINEAR PROBLEM. Au -

(

2x2y2 + 2x2y + 2xy2 - 6xy)e(Z+Y)

>

t&n The analytical

solution

= 0,

in R = (0,l)

= 0.

x (0, l), (60)

is ueXact = x(x - l)y(y

- l)e(“+“).

Numerical results are presented in Figure 3a. We do not have an explanation PDECB solution is more accurate than the interpolation. EXAMPLE 3. 1D GELFAND-BRATU

as to why the MQ

PROBLEM. This is a scalar problem

u” + Xe” = 0,

u(0) = u(1) = 0

in fi = (0, l),

(61)

A. I. FEDOSEYEV et al.

450

1D PDE BY PDECB AND KANSA-MQ

1e-06

le-07

10-08. 4

6

8

12

10

14

16

l/h

18

20

22

(a) 1D linear PDE solution. (a) 1D linear problem, equation (59); the L, norm of the solution error is plotted, in the logarithmic scale, versus l/h, where h is the average distance between the nodes. The roundoff error starts to dominate at l/h z 11 for KansaMQ method and at l/h M 18 for the MQ PDECB method. 1D BRATU-GELFAND EQUATION BY PDECB

0.1

MO PDECB-1 D 4-. K-MQ .n.. 0.01

0.001

5

5

.I\

x .*

‘\

..

‘3 \

....

m

‘,

o.ooo1

a t I

i

..... ‘.

..

‘,

‘\

‘\

le-05

‘.

*.

*.

x.

-*._. ..

‘.

“.x.._ ‘\

f u!

%_

“9

1e-06

‘\

.... ‘.

.... ‘.

‘\

t

xx

‘,

“4‘..\

1e-07

‘..\

‘..\ le-08

‘T, ‘.

l\ ‘.

1e-09

le-10 L 2

4

6

1%

10

12

‘.\

m 14

(b) 1D continuation. (b) The location X of the limit point for 1D Bratu-Gelfand problem, equation (61). Relative error in X is plotted in the logarithmic scale versus l/h. Figure 2. method.

Convergence

properties

of the Kansa-MQ

method

and the MQ PDECB

that appears in combustion theory and is used as the demo example exp in AUTO97 [IS] (fifthorder adaptive orthogonal spline collocation method). There is a limit (fold) point on the solution

Improved Multiquadric

Method

451

2D MQ INTERPOLATION, AND SOLUTION BY PDECB, KANSA-MQ PDECB %K-MQ -t--. MQ INTERP. -a-. 0.1

25

0.01

iz p 3

0.001

ci

’

0.0001

I

1e-05 2

4

6

I

I

8

10

12

14

16

I/h (a) 2D linear PDE solution and interpolation. (a) 2D linear PDE, equation (60); the I&,-norm of the solution error is plotted, in the logarithmic scale, versus l/h, where h is the average distance between the nodes. The roundoff error starts to dominate at l/h N 9 for the Kansa-MQ method and at l/h N 11 for the MQ PDECB method. We also provide, for comparison, the error in the MQ interpolation of the exact solution uexact. 1D BRATU-GELFAND EQUATION

0.1

. PDECB 4 Kansa-MQ -I+-.

0.01

g

0.001

Li 0.0001

;

E 1 e-05

1e-06 5

6

7

l:h

9

10

11

(b) 2D continuation. (b) The location X of the limit point for 2D Bratu-Gelfand problem, equation (62). Relative error in X is plotted, in the logarithmic scale, versus l/h. Figure 3. Convergence

properties

of the Kansa-MQ

and the MQ PDECB

methods.

curve. We take the value of X at the limit point found from demo exp (K 2 50) as exact. The relative error in location of the limit point is shown in Figure 2b. See also [I] for additional numerical results and references.

A. I.

452

EXAMPLE 4. 2D GELFAND-BRATU

FEDOSEYEV et al.

PROBLEM.

Au + Xe” = 0,

in R = (0,l)

x (0, l), (62)

ulan = 0.

This

problem

a high-order point

was studied orthogonal

on the solution

of X obtained of the limit obtained

curve.

of authors.

collocation

method

The exact location

In [26], the problem with sparse Jacobian.

of the limit point

in [26] on a 16 x 16 mesh with 4 x 4 collocation point

is shown

in Figure

[l] using

quadruple

precision

use only double precision results,

by a number

spline

references,

3b.

Note that method

of the operation

is assumed The relative

is a limit

with (fold)

to be at the value error in location

the curve for the Kansa-MQ

which considerably

with the MQ PDECB

and a discussion

points.

was discretized There

method

slowed down computations,

here. See also [l] for additional

was

while we numerical

count.

EXAMPLE 5. 1D MODEL LINEAR SINGULAR PERTURBATION PROBLEM. EU,,

+

in fi = (0, l),

uz = 0,

u(0) = 0,

The analytical

solution

u(l)

(63)

= 1.

is (1 - em”/‘) U

cG3ct

=

(1

_

e-1/e)

.

This problem was studied by Hon in [27], who found that a standard Kansa-MQ crude in the case E