Weighted-Averaging Finite-Element Method for 2D Elastic Wave Equations in the Frequency Domain

2003 ◽  
Vol 93 (2) ◽  
pp. 904-921 ◽  
Author(s):  
D.-J. Min
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elastic Wave ◽  
Frequency Domain ◽  
Wave Equations ◽  
Weighted Averaging ◽  
Elastic Wave Equations ◽  
Element Method

Accuracy of finite‐difference and finite‐element modeling of the scalar and elastic wave equations

Geophysics ◽  
1984 ◽  
Vol 49 (5) ◽  
pp. 533-549 ◽  
Author(s):  
Kurt J. Marfurt
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Elastic Wave ◽  
Frequency Domain ◽  
Poisson’S Ratio ◽  
Wave Equations ◽  
Cost Effective ◽  
Poisson's Ratio ◽  
Solution Technique ◽  
Elastic Wave Equations

Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite‐difference and finite‐element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time‐domain and frequency‐domain techniques. In addition to versatility and ease of implementation, it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors and transmission coefficients. This paper shows that in homogeneous media the explicit finite‐element and finite‐difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson’s ratio less than 0.3. Finite‐elements are superior to finite‐differences when modeling elastic media with Poisson’s ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central‐differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency‐domain finite‐element solutions employing a weighted average of consistent and lumped masses yield the most accurate results, and they promise to be the most cost‐effective method for CDP, well log, and interactive modeling.

scholarly journals A GALERKIN FINITE ELEMENT METHOD TO SOLVE FRACTIONAL DIFFUSION AND FRACTIONAL DIFFUSION-WAVE EQUATIONS

2013 ◽  
Vol 18 (2) ◽  
pp. 260-273 ◽  
Author(s):  
Alaattin Esen ◽  
Yusuf Ucar ◽  
Nuri Yagmurlu ◽  
Orkun Tasbozan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Fractional Diffusion ◽  
Galerkin Finite Element Method ◽  
Diffusion Wave ◽  
Galerkin Finite Element ◽  
Base Functions ◽  
Element Method

In the present study, numerical solutions of the fractional diffusion and fractional diffusion-wave equations where fractional derivatives are considered in the Caputo sense have been obtained by a Galerkin finite element method using quadratic B-spline base functions. For the fractional diffusion equation, the L1 discretizaton formula is applied, whereas the L2 discretizaton formula is applied for the fractional diffusion-wave equation. The error norms L 2 and L ∞ are computed to test the accuracy of the proposed method. It is shown that the present scheme is unconditionally stable by applying a stability analysis to the approximation obtained by the proposed scheme.

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