TY - JOUR
T1 - An a Priori Error Analysis of Operator Upscaling for the Acoustic Wave Equation
AU - Vdovina , T.
AU - Minkoff , S. E.
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 543
EP - 569
PY - 2008
DA - 2008/05
SN - 5
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/826.html
KW - upscaling, convergence analysis, acoustic wave propagation, multiscale methods, error estimates.
AB - <p style="text-align: justify;">In many earth science problems, the scales of interest range
from centimeters to kilometers. Computer power and time limitations
prevent inclusion of all the fine-scale features in most models.
However, upscaling methods allow creation of physically realistic and
computationally feasible models. Instead of solving the problem
completely on the fine scale, upscaling methods produce a coarse-scale
solution that includes some of the fine-scale detail. Operator-based
upscaling applied to the pressure/acceleration formulation of the
acoustic wave equation solves the problem via decomposition of the
solution into coarse and subgrid pieces. To capture local fine-scale
information, small subgrid problems are solved independently in each
coarse block. Then these local subgrid solutions are included in the
definition of the coarse problem. In this paper, accuracy of the
upscaled solution is determined via a detailed finite element analysis
of the continuous-in-time and fully-discrete two-scale numerical
schemes. We use lowest-order Raviart-Thomas mixed finite element
approximation spaces on both the coarse and fine scales. Energy
techniques show that in the $L^2$ norm the upscaled acceleration converges
linearly on the coarse scale, and pressure (which is not upscaled in
this implementation) converges linearly on the fine scale. The fully
discrete scheme is also shown to be second-order in time. Three
numerical experiments confirm the theoretical rate of convergence results.</p>