Finite Difference Approximation of the Weak Solution of a Mildly Nonlinear Dirichlet Problem

1973 ◽  
Vol 10 (4) ◽  
pp. 635-646
Author(s):  
M. B. Rosenzweig
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Finite Difference ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Nonlinear Dirichlet Problem

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 Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters

2021 ◽  
Author(s):  
Tesfaye Aga Bullo ◽  
Guy Aymard Degla ◽  
Gemechis File Duressa
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Variable Parameter ◽  
Accurate Solution ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Parabolic Problems ◽  
Difference Approximation ◽  
Uniform Mesh

A parameter-uniform finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with two parameters. The solution involves boundary layers at both the left and right ends of the solution domain. A numerical algorithm is formulated based on uniform mesh finite difference approximation for time variable and appropriate piecewise uniform mesh for the spatial variable. Parameter-uniform error bounds are established for both theoretical and experimental results and observed that the scheme is second-order convergent. Furthermore, the present method produces a more accurate solution than some methods existing in the literature.   

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