introduction
references
chapter 1 global abcs for second order elliptic equations
1.1 exterior problem of second order elliptic equations
1.2 global abcs for the exterior problem of 2-d poisson
equation
1.2.1 steklov-poincaré mapping for the exterior problem of laplace
equation
1.2.2 the reduced boundary value problem oni.
1.2.3 finite element approximation of the reduced boundary value
problem 1.2.30~1.2.32
1.3 global abcs for the exterior problems of 3-d poisson
equation
1.3.1 exact and approximate abcs on the spherical artificial
boundary γr
1.3.2 equivalent and approximate boundary value problems on the
bounded computational domaini
1.3.3 finite element approximation of the variational problem
1.3.30
1.4 exterior problem of the modified helmholtz equation
1.4.1 global boundary condition of the exterior problem for the 2-d
modified helmholtz equation
1.4.2 the reduced boundary value problem on the computational
domaini
1.4.3 finite element approximation of the reduced boundary value
problem
1.4.4 global boundary condition of the exterior problem for the 3-d
modified helmholtz equation
1.5 global abcs for the exterior problems of the helmholtz
equation
1.5.1 dirichlet to sommerfeld mapping of the exterior problem of
the 2-d helmholtz equation
1.5.2 dirichlet to sommerfeld mapping of the exterior problem of
the 3-d helmholtz equation
references
chapter 2 global abcs for the navier system and stokes
system
2.1 navier system and stokes system
2.2 the exterior problem of the 2-d navier system
2.2.1 the global boundary condition on the artificial boundary
γr
2.2.2 the reduced problem on the bounded domain
2.2.3 the finite element approximation for the reduced problem
2.2.59
2.3 exterior problem of the 2-d stokes system
2.3.1 highly accurate approximate artificial boundary
condition
2.3.2 finite element approximation on the computational domaini for
the reduced problem
2.4 vector fields on the spherical surface.
2.5 global abcs for the exterior problem of 3-d navier
system.
2.5.1 highly accurate approximate abcs
2.5.2 finite element approximation of the variational problem on
the bounded computational domaini 100 references
chapter 3 global abcs for heat and schr.dinger equations
3.1 heat equations on unbounded domains
3.2 1-d heat equations on unbounded domains
3.2.1 exact boundary conditions on the artificial boundary σ
3.2.2 finite difference approximation for the reduced problem
3.2.7~3.2.10
3.2.3 stability analysis of scheme 3.2.29~3.2.33
3.3 global boundary conditions for exterior problems of 2-d heat
equations
3.3.1 exact and approximate conditions on the artificial boundary
σr.
3.3.2 finite difference approximation of the reduced problem
3.3.37~3.3.40
3.4 global boundary conditions for exterior problems of 3-d heat
equations
3.4.1 exact and approximate conditions on the artificial boundary
σr.
3.4.2 stability analysis for the reduced initial boundary value
problem
3.4.3 the finite element approximation for the reduced initial
boundary value problem 3.4.38~3.4.41
3.5 schr.dinger equation on unbounded domains
3.6 1-d schr.dinger equation on unbounded domains.
3.6.1 the reduced initial value problem and its finite difference
approximation
3.6.2 stability and convergence analysis of scheme
3.6.19~3.6.22
3.7 the global boundary condition for the exterior problem of the
2-d linear schr.dinger equation
3.7.1 exact and approximate boundary conditions on the artificial
boundary σr
3.7.2 stability analysis of the reduced approximate initial
boundary value problem
3.8 the global boundary condition for the exterior problem of the
3-d linear schr.dinger equation
3.8.1 exact and approximate boundary conditions on the artificial
boundary σr
3.8.2 stability analysis of the reduced approximate initial
boundary value problem
references
chapter 4 abcs for wave equation, klein-gordon equation, and
linear kdv equations
4.1 1-d wave equation
4.1.1 transparent boundary conditions on the artificial boundaries
σ1 and σ
4.2 2-d wave equation
4.2.1 absorbing boundary conditions
4.2.2 the initial boundary value problem on the bounded
computational domain di
4.3 3-d wave equation
4.3.1 absorbing boundary condition on the artificial boundary
σr
4.3.2 the equivalent and approximate initial boundary value problem
on the bounded computational domain di
4.4 1-d klein-gordon equation
4.4.1 absorbing boundary conditions on the artificial boundary σ1,
σ
4.4.2 the initial boundary value problem on the bounded
computational domain di
4.5 2- and 3-d klein-gordon equations.
4.5.1 absorbing boundary conditions on the artificial boundary σr
2-d case
4.5.2 absorbing boundary conditions on the artificial boundary σr
3-d case
4.5.3 the initial boundary value problem on the bounded
computational domain di
4.6 linear kdv equation
4.6.1 absorbing boundary condition on the artificial boundaries σa
and σb
4.6.2 the equivalent initial boundary value problem on the bounded
computational domain
4.7 appendix: three integration formulas
references
chapter 5 local artificial boundary conditions
5.1 local boundary conditions for exterior problems of the 2-d
poisson equation
5.1.1 local boundary condition on the artificial bboundary γr
5.1.2 finite element approximation using the local boundary
condition and its error estimate
5.2 local boundary conditions for the 3-d poisson equation
5.2.1 the local boundary condition on the artificial boundary γr
for problem i
5.2.2 local boundary conditions on the artificial boundary γr for
problem ii
5.3 local abcs for wave equations on unbounded domains
references
chapter 6 discrete artificial boundary conditions
6.1 boundary condition on a polygon boundary for the 2-d poisson
equation—the method of lines
6.1.1 discrete boundary conditions on polygonal boundaries
6.1.2 numerical approximation of the exterior problem
6.1.1~6.1.3
6.2 2-d viscous incompressible flow in a channel—infinite
difference method
6.2.1 2-d viscous incompressible flow in a channel
6.2.2 discrete abcs
6.3 numerical simulation of infinite elastic foundation—infinite
element method
6.3.1 the steklov-poincarè on an artificial boundary of line
segments
6.3.2 numerical approximation for the bilinear form bu, v
6.3.3 a direct method for solving the infinite system of algebraic
equations 6.3.25
6.3.4 a fast iteration method for computing the combined stiffness
matrix kz.
6.4 discrete absorbing boundary condition for the 1-d klein-gordon
equation—z transform method
6.4.1 z transform
6.4.2 discrete absorbing abc
6.4.3 finite difference approximation for the 1-d klein-gordon
equation on the bounded domain.296 references
chapter 7 implicit artificial boundary conditions
7.1 implicit boundary condition for the exterior problem of the 2-d
poisson equation
7.1.1 the single and double layer potential, and their derivative
for the 2-d laplace equation
7.1.2 the derivation of the implicit abc for the exterior problem
of the 2-d poisson equation
7.1.3 the finite element approximation and error estimate for the
variational problem 7.1.37
7.2 implicit boundary condition for the exterior problem of the 3-d
poisson equation
7.3 abc for the exteriorproblem of the helmholtz equation
7.3.1 the normal derivative on γa for the double layer potential of
the helmholtz equation
7.4 implicit abcs for the exterior problems of the navier
system.
7.4.1 fundamental solution, stress operator, single and double
layer potentials
7.4.2 new forms of t.x, nxvii x on γa n = 2
7.4.3 new forms of t.x, nxvii x on γa n = 3
7.4.4 implicit abc for the exterior problem
7.5 implicit abcs for the sound wave equation.
7.5.1 the kirchhoff formula for the 3-d sound wave equation
references
chapter 8 nonlinear artificial boundary conditions
8.1 the burgers equation
8.1.1 nonlinear abcs for the burgers equation
8.1.2 the equivalent initial boundary value problem on the bounded
computational domain di
8.2 the kardar-parisi-zhang equation
8.2.1 nonlinear abc for the k-p-z equation d = 1
8.2.2 nonlinear abc for the k-p-z equation d = 2
8.2.3 nonlinear abc for the k-p-z equation d = 3
8.3 the cubic nonlinear schr.dinger equation.
8.3.1 nonlinear boundary conditions on the artificial boundaries σ0
and σ.
8.3.2 the equivalent initial boundary value problem on the bounded
domain [–1, 0] × [0, t ]
8.4 operator splitting method for constructing approximate
nonlinear abcs
8.4.1 the local absorbing abc for the linear schr.dinger
equation
8.4.2 finite difference approximation on the bounded computational
domain.360 references
chapter 9 applications to problems with singularity
9.1 the modified helmholtz equation with a singularity
9.1.1 abc near singular points
9.1.2 an iteration method based on the abc
9.2 the interface problem with a singularity
9.2.1 a discrete boundary condition on the artificial boundary
γr
9.2.2 finite element approximation
9.3 the linearelastic problem with asingularity
9.3.1 discrete boundary condition on the artificial boundary
γr
9.3.2 an iteration method based on the abc
9.4 the stokes equations with a singularity
9.4.1 the discrete boundary condition on the artificial boundary
γr
9.4.2 singular finite element approximation
references
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