1 Introduction
The boundary element method (BEM) is the most potent tool for the
solution of linear elliptic partial differential equations (PDEs)
in an exterior domain. The standard
BEM also serves as an attractive alternative to methods such as
the finite element and finite difference methods for interior
linear elliptic problems. The BEM has thus become an established
computational method over recent years (Jaswon and Symm (1977),
Brebbia (1978), Banerjee and Butterfield (1981)). However, apart from
suffering the limitation of being inappropriate for non-linear
problems, the BEM is unable to cope directly
with discontinuities in the variables
in the domain of the PDE. The discontinuity will be assumed to
have the topology of a shell - an open surface in three-dimensional
problems, a line in two dimensions. An illustration
of the general domain is given in figure 1.
The traditional BEM is derived from a boundary integral equation
(BIE) formulation of the PDE by dividing the
boundary into boundary elements and applying an integral
equation method (usually collocation) to obtain the method of solution.
For domains in the form of figure 1, the traditional BEM can be applied
by subdividing the domain into subdomains, as illustrated in figure 2.
Boundary integral equation reformulations of the PDEs on
each subdomain can now
be obtained and, through coupling these equations
across common boundaries, the solution
throughout the domain can be obtained.
A similar method to the traditional BEM
for the solution of a PDE in the infinite domain exterior to a shell
discontinuity can be derived through recasting the PDE as an integral equation
termed a shell integral equation (SIE). A numerical method
can then be derived in a similar way to the BEM (see, for example,
Ben Mariem and Hamdi (1987) or Warham (1988)).
However, the main value of shell elements is in their use in conjunction with
boundary elements. In this paper, it is shown how the PDE in the domain of figure
1 can be reformulated straightforwardly as an integral equation termed
a boundary and shell integral equation (BSIE) and thus how a
numerical method termed the boundary and shell element method (BSEM)
can be derived.
Boundary element methods have traditionally fallen into two distinct classes,
direct BEMs and indirect BEMs, based on direct and indirect integral
equation formulations.
In this paper direct and indirect boundary and shell integral equation
formulations are given for the interior two-dimensional Laplace equation.
The BSIEs are a hybrid of the corresponding direct or
indirect boundary integral equation with the shell integral equation.
Hence new integral equation based methods for the solution
of linear elliptic PDEs with discontinuities
in its domain are introduced in this paper. The methods are demonstrated on the
two-dimensional interior Laplace problem
An illustration of the domain is given in figure 1. It consists
of a region D with boundary S and with shell discontinuities
G.
In order to specify the problem fully a conditions for points on the
boundary and
on the shell must be stated. These are the boundary condition and
the shell condition.
In Warham (1988), the shell integral equations are derived by first
assuming that the shells have finite thickness and hence the standard
boundary integral equation formulation is valid. The shell thickness
is then allowed to approach zero. A similar limiting process can
be used to derive the boundary and shell integral equation
by assuming S to be fixed and taking the limit as the thickness
of the shells approach zero.
The integral equation formulations of the interior Laplace problem
are stated in section 2.
In order to derive a particular method, the boundary and
shell are divided into uniform elements and the functions defined on the
boundary and shell are approximated by a constant on each element.
The integral equation method is then derived through collocation.
The methods are demonstrated through their application to a
test problem where the domain is the unit square and a discontinuity lies
between ([1/2],[1/2]) and ([1/2],1).