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Advanced Scientific Computing
Comparison of Two Methods for Elliptic Boundary Value Problems
J. Du, S. Wang, J. Glimm, and R. Samulyak
The purpose of this study is to present a comparison of two popular
methods for the solution of elliptic boundary value problems: the
embedded boundary method and the mixed finite element methods. The
methods are quite different in nature and performance characteristics.
There are many methods for solving the elliptic boundary value/interface
problems. Several popular methods have been developed on cartesian
meshes for the boundary value/interface problems: the immersed boundary
method (IBM) by Perskin [1], the immersed interface method by LeVeque
and Li [2], the embedded boundary method by Johansen and Colella [3].
The advantage of these methods is that they are defined on a cartesian
grid and there is no need of mesh generation. For the cells away from
the boundary/interface, they just use a central finite difference
method, which is simple and second order accurate. For the cells that
intersect with the boundary/interface, special treatment is needed.
Another set of approaches is finite element methods with unstructured
mesh, which is more suitable for complex geometries.
We investigate two methods for the elliptic boundary value problem: one
is the embedded boundary method (EBM) and the other is the mixed-hybrid
finite element method (MFM) [4] with unstructured grid using different
basis functions (RT0, BDM1 and RT1). The computational domain and the
unstructured mesh are shown in Figure 1. The exponential function is
used as the solution of the elliptic problem. Parallel iterative solvers
based on PETSc are used for solving the discretized linear system. We
compare both the solution errors and performances of the two different
methods (Table 1 and Table 2) [5]. Figure 2 shows the L2 error of the
solution gradient.
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Figure 1. Computational domain and
unstructured grid. |
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Figure 2. L2 error of solution gradient
for EBM (left) and RT1 (right) for 128x128 grid. |
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Table 1. Maximum gradient errors on domain boundary. |
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Table 2. Convergence and timing comparisons. |
From these results, we
draw the following conclusions. Since the embedded boundary method uses a
structured cartesian grid, it is relatively easy to implement while it is
harder to write the mesh generation program. But after the mesh is given,
the discretization is simpler for the mixed finite element method. And it is
easier to use the mixed finite element for the elliptic interface problem
since the interface is in fact an internal boundary. However, the EBM method
must be modified to solve such problems. Also FEM could have higher accuracy
if high order basis function is used. To save computational resources when
solving large problems, we could use EBM with automatic mesh refinement
since it has better performance for the same order of accuracies.
The EBM has the advantage of fewer unknowns with the same accuracy compared
with the MFM. There are two reasons for this. One reason is that the EBM
uses a structured grid and the finite volume/central finite difference has
super convergence in the mesh. The MFM uses an unstructured grid, and to
achieve the same order of accuracy, a higher order basis function space is
needed, which means more unknowns. The other reason is that the unknowns for
EBM are cell centered and those for the MFM are edge centered. Since the
approximate ratio of the vertices to faces to edges is 1:2:3 for a simple
large triangle mesh, we know the ratio of the unknowns for the EBM, RT0,
BDM1, RT1 is approximately 1:3:6:6. Thus the EBM problem is smaller, which
explains why it is much faster.
References
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[1] Perskin, C.S. The immersed boundary method. Acta Numerica 11:
479-517 (2002).
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[2] LeVeque, R.J. and Li, Z.L. The immersed interface method for
elliptic equations with discontinuous coefficients and singular sources.
SIAM J. Numer. Anal. 31: 1019-1044 (1994).
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[3] Johansen, H. and Colella, P. A Cartesian grid embedding boundary
method for Poisson's equation on irregular domains. J. Comput. Phys.
147: 60-85 (1998).
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[4] Chavent, G. and Roberts, J.E. A unified physical presentation of
mixed, mixed-hybrid finite elements and standard finite difference
approximations for the determination of velocities in waterflow
problems. Adv. Water Resources 14(6): 329-348 (1991).
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[5] Du, J., Wang, S., Glimm, J., and Samulyak, R. A comparison study of
two methods for elliptic boundary value problems. SIAM J. Sci.
Computing. Submitted, 2006.
Last Modified: January 31, 2008
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