scholarly journals Finite difference approximation of an elliptic problem with nonlocal boundary condition

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1549-1557
Author(s):  
Sandra Hodzic ◽  
Bosko Jovanovic
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Finite Difference ◽  
Existence And Uniqueness ◽  
Input Data ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Sobolev Norm ◽  
Finite Difference Approximation ◽  
Difference Approximation

We consider Poisson?s equation on the unit square with a nonlocal boundary condition. The existence and uniqueness of its weak solution in Sobolev spaceH1 is proved. A finite difference scheme approximating this problem is proposed. An error estimate compatible with the smoothness of input data in discrete H1 Sobolev norm is obtained.

scholarly journals Investigation of complex eigenvalues for a stationary problem with two-point nonlocal boundary condition

2011 ◽  
Vol 52 ◽  
pp. 303-308
Author(s):  
Kristina Skučaitė-Bingelė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Stationary Problem ◽  
Finite Difference Schemes ◽  
Complex Eigenvalues ◽  
Sturm Liouville

The Sturm–Liouville problem with one classical and another two-point nonlocal boundary condition is considered in this paper. These problems with nonlocal boundary condition are not self-adjoint, so the spectrum has complex points. We investigate how the spectrum in the complex plane of these problems (and for the Finite-Difference Schemes) depends on parameters γ  and ξ  of the nonlocal boundary conditions.

scholarly journals Investigation of the Sturm–Liouville problems with integral boundary condition

2011 ◽  
Vol 52 ◽  
pp. 297-302
Author(s):  
Agnė Skučaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Trapezoidal Rule ◽  
Integral Conditions ◽  
Integral Type ◽  
Integral Boundary ◽  
Complex Eigenvalues ◽  
Sturm Liouville

This paper presents some new results on the spectrum of a complex plane for the second order Finite-Difference Scheme with one integral type nonlocal boundary condition (NBC). We analyze how complex eigenvalues of these problems depend on the parameters of the integral NBC. The integral conditions are approximated by the trapezoidal rule or by Simpson’s rule.

A transparent boundary for finite‐difference wave simulation

Geophysics ◽  
1990 ◽  
Vol 55 (2) ◽  
pp. 201-208 ◽  
Author(s):  
L. T. Long ◽  
J. S. Liow
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Grid Point ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Angle Of Incidence ◽  
P Waves ◽  
Difference Wave ◽  
Transparent Boundary Condition ◽  
Wave Simulation

Transparent (or absorbing) boundaries can be used in finite‐difference wave simulation to reduce the size of the computational grid and to eliminate reflections from the edges. An efficient and accurate transparent boundary can be formulated by decomposing the elastic waves into dilatational and rotational strains. The wave motions for the strains at the boundary can then be approximated by a one‐way wave equation. The direction of propagation is determined at each grid point by the gradient. This transparent boundary condition eliminates artificial reflections for a wave arriving at any angle of incidence and reduces the error to the level of precision of the finite‐difference approximation. Application of this transparent boundary condition is restricted to a medium that is homogeneous at the boundary to assure full separation of P waves from S waves. Also, interfering waves that generate phase velocities significantly greater than the assumed group velocity introduce errors. An example of the transparent boundary condition shows that it is a significant improvement over the A1 boundary condition of Clayton and Engquist.

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