Most recently, we have developed a novel multilevel boundary element method (MLBEM) for steady Stokes flows in irregular two-dimensional domains (Grigoriev, M.M., and Dargush, G.F., Comput. Methods. Appl. Mech. Eng., 2005). The multilevel algorithm permitted boundary element solutions with slightly over 16,000 degrees of freedom, for which approximately 40-fold speedups were demonstrated for the fast MLBEM algorithm compared to a conventional Gauss elimination approach. Meanwhile, the sevenfold memory savings were attained for the fast algorithm. This paper extends the MLBEM methodology to dramatically improve the performance of the original multilevel formulation for the steady Stokes flows. For a model problem in an irregular pentagon, we demonstrate that the new MLBEM formulation reduces the CPU times by a factor of nearly 700,000. Meanwhile, the memory requirements are reduced more than 16,000 times. These superior run-time and memory reductions compared to regular boundary element methods are achieved while preserving the accuracy of the boundary element solution.

1.
Barnes
,
J.
, and
Hut
,
P.
, 1986, “
A Hierarchical Force Calculation Algorithm
,”
Nature (London)
0028-0836
324
, pp.
446
449
.
2.
Grama
,
A.
,
Kumar
,
A.
, and
Sameh
,
A.
, 1998, “
Parallel Hierarchical Solvers and Preconditioners for Boundary Element Methods
,”
SIAM J. Sci. Comput. (USA)
1064-8275
20
, pp.
337
358
.
3.
Beylkin
,
G.
,
Coifman
,
R.
, and
Rokhlin
,
V.
, 1991, “
Fast Wavelet Transforms and Numerical Algorithms: 1
,”
Commun. Pure Appl. Math.
0010-3640
44
, pp.
141
183
.
4.
Alpert
,
B.
,
Beylkin
,
G.
,
Coifman
,
R.
, and
Rokhlin
,
V.
, 1993, “
Waveletlike Bases for the Fast Solution of Second-Kind Integral Equations
,”
SIAM J. Sci. Comput. (USA)
1064-8275
14
, pp.
159
184
.
5.
Rokhlin
,
V.
, 1985, “
Rapid Solution of Integral Equations of Classical Potential Theory
,”
J. Comput. Phys.
0021-9991
60
, pp.
187
207
.
6.
Greengard
,
L.
, and
Rokhlin
,
V.
, 1987, “
A Fast Algorithm for Particle Simulations
,”
J. Comput. Phys.
0021-9991
73
, pp.
325
348
.
7.
Nabors
,
K.
,
Korsmeyer
,
F. T.
,
Leighton
,
F. T.
, and
White
,
J.
, 1994, “
Preconditioned, Adaptive, Multipole-Accelerated Iterative Methods for Three-Dimensional First-Kind Integral Equations of Potential Theory
,”
SIAM J. Sci. Comput. (USA)
1064-8275
15
, pp.
713
735
.
8.
Rahola
,
J.
, 1996, “
Diagonal Forms of the Translation Operators in the Fast Multipole Algorithm for Scattering Problems
,”
BIT
0006-3835
36
, pp.
333
358
.
9.
Cheng
,
H.
,
Greengard
,
L.
, and
Rokhlin
,
V.
, 1999, “
A Fast Adaptive Multipole Algorithm in Three Dimensions
,”
J. Comput. Phys.
0021-9991
155
, pp.
468
498
.
10.
Koc
,
S.
,
Song
,
J.
, and
Chew
,
W. C.
, 1999, “
Error Analysis for the Numerical Evaluation of the Diagonal Forms of the Scalar Spherical Addition Theorem
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
0036-1429
36
, pp.
906
921
.
11.
Darve
,
E.
, 2000, “
The Fast Multipole Method: Numerical Implementation
,
J. Comput. Phys.
0021-9991
160
, pp.
195
240
.
12.
Darve
,
E.
, 2000, “
The Fast Multipole Method 1: Error Analysis and Asymptotic Complexity
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
0036-1429,
38
, pp.
98
-
128
.
13.
Brandt
,
A.
, and
Lubrecht
,
A. A.
, 1990, “
Multilevel Matrix Multiplication and Fast Solution of Integral Equations
,”
J. Comput. Phys.
0021-9991
90
, pp.
348
370
.
14.
Lubrecht
,
A.
, and
Ioannides
,
E.
, 1991, “
A Fast Solution of the Dry Contact Problem and the Associated Subsurface Stress Field Using Multilevel Techniques
,”
ASME J. Tribol.
0742-4787
113
, pp.
128
133
.
15.
Grigoriev
,
M. M.
, and
Dargush
,
G. F.
, 2004, “
A Multi-Level Boundary Element Method for Two-Dimensional Steady Heat Diffusion
,”
Numer. Heat Transfer, Part B
1040-7790
46
, pp.
329
356
.
16.
Grigoriev
,
M. M.
, and
Dargush
,
G. F.
, 2004, “
A Fast Multi-Level Boundary Element Method for the Helmholtz Equation
,”
Comput. Methods Appl. Mech. Eng.
0045-7825
193
, pp.
165
203
.
17.
Wang
,
C. H.
,
Grigoriev
,
M. M.
, and
Dargush
,
G. F.
, 2005, “
A Fast Multi-Level Convolution Boundary Element Method for Transient Diffusion Problems
,”
Int. J. Numer. Methods Eng.
0029-5981
62
, pp.
1895
1926
.
18.
Polonsky
,
I. A.
, and
Keer
,
L. M.
, 1999, “
A Numerical Method for Solving Rough Contact Problems Based on the Multi-Level Multi-Summation and Conjugate Gradient Techniques
,”
Wear
0043-1648
231
, pp.
206
219
.
19.
Grigoriev
,
M. M.
, and
Dargush
,
G. F.
, 2005, “
A Multi-Level Boundary Element Method for Stokes Flows in Irregular Two-Dimensional Domains
,”
Comput. Methods Appl. Mech. Eng.
0045-7825
194
, pp.
3553
3581
.
20.
Polonsky
,
I. A.
, and
Keer
,
L. M.
, 2000, “
Fast Methods for Solving Rough Contact Problems: A Comparative Study
,
ASME J. Tribol.
0742-4787
122
, pp.
36
41
.
21.
Venner
,
C. H.
, and
Lubrecht
,
A. A.
, 2000,
Multi-level Methods in Lubrication
,
Elsevier
, Amsterdam.
22.
Brandt
,
A.
, 1977, “
Multi-level Adaptive Solutions to Boundary-Value Problems
,”
Math. Comput.
0025-5718
31
, pp.
333
390
.
23.
Costabel
,
M.
, 1988, “
Boundary Integral Operators on Lipschitz Domains: Elementary Results
,”
SIAM J. Math. Anal.
0036-1410,
19
, pp.
613
626
.
24.
Laubin
,
P.
, 2001, “
Optimal Order Collocation for the Mixed Boundary Value Problem on Polygons
,”
Math. Comput.
0025-5718
70
, pp.
607
636
.
25.
Press
,
W. H.
,
Teukolsky
,
S. A.
,
Vetterling
,
W. T.
, and
Flannery
,
B. P.
, 1992,
Numerical Recipes in C, The Art of Scientific Computing
,
Cambridge University Press
, Cambridge.
You do not currently have access to this content.