An a Priori Error Analysis of Operator Upscaling for the Acoustic Wave Equation

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 L2 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.