scholarly journals Finite-difference approximation of free-discontinuity problems

2001 ◽  
Vol 131 (3) ◽  
pp. 567-595 ◽  
Author(s):  
Massimo Gobbino ◽  
Maria Giovanna Mora
Keyword(s):  
Finite Difference ◽  
Finite Differences ◽  
Pointwise Convergence ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Compactness Result ◽  
Free Discontinuity Problems ◽  
Non Local ◽  
Γ Convergence

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence and a compactness result, which implies, in particular, the convergence of minima and minimizers.

scholarly journals Finite-difference approximation of free-discontinuity problems

2001 ◽  
Vol 131 (3) ◽  
pp. 567-595
Author(s):  
Massimo Gobbino ◽  
Maria Giovanna Mora
Keyword(s):  
Finite Difference ◽  
Finite Differences ◽  
Pointwise Convergence ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Compactness Result ◽  
Free Discontinuity Problems ◽  
Non Local ◽  
Γ Convergence

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence and a compactness result, which implies, in particular, the convergence of minima and minimizers.

Finite differences on minimal grids

Geophysics ◽  
1994 ◽  
Vol 59 (9) ◽  
pp. 1435-1443 ◽  
Author(s):  
Sophie‐Adélade Magnier ◽  
Peter Mora ◽  
Albert Tarantola
Keyword(s):  
Finite Difference ◽  
Conventional Method ◽  
Finite Differences ◽  
Quantitative Estimation ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Elastic Media ◽  
Numerical Tests ◽  
Propagation Of Waves

Conventional approximations to space derivatives by finite differences use orthogonal grids. To compute second‐order space derivatives in a given direction, two points are used. Thus, 2N points are required in a space of dimension N; however, a centered finite‐difference approximation to a second‐order derivative may be obtained using only three points in 2-D (the vertices of a triangle), four points in 3-D (the vertices of a tetrahedron), and in general, N + 1 points in a space of dimension N. A grid using N + 1 points to compute derivatives is called minimal. The use of minimal grids does not introduce any complication in programming and suppresses some artifacts of the nonminimal grids. For instance, the well‐known decoupling between different subgrids for isotropic elastic media does not happen when using minimal grids because all the components of a given tensor (e.g., displacement or stress) are known at the same points. Some numerical tests in 2-D show that the propagation of waves is as accurate as when performed with conventional grids. Although this method may have less intrinsic anisotropies than the conventional method, no attempt has yet been made to obtain a quantitative estimation.

scholarly journals NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS IN IRREGULAR PLANE REGIONS USING QUALITY STRUCTURED CONVEX GRIDS

2013 ◽  
Vol 04 (02) ◽  
pp. 1350004
Author(s):  
F. DOMÍNGUEZ-MOTA ◽  
P. FERNÁNDEZ-VALDEZ ◽  
S. MENDOZA-ARMENTA ◽  
G. TINOCO-GUERRERO ◽  
J. G. TINOCO-RUIZ
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Numerical Solution ◽  
Poisson Equation ◽  
Finite Differences ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Structured Grids ◽  
Simply Connected ◽  
Simply Connected Domains

The variational grid generation method is a powerful tool for generating structured convex grids on irregular simply connected domains whose boundary is a polygonal Jordan curve. Several examples that show the accuracy of a finite difference approximation to the solution of a Poisson equation using this kind of structured grids have been recently reported. In this paper, we compare the accuracy of the numerical solution calculated using those structured grids and finite differences against the solution obtained with Delaunay-like triangulations on irregular regions.

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.

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