Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. MULTISCALE MODEL.SIMUL. c \ 2020 Society for Industrial and Applied Mathematics Vol. 18, No. 2, pp. 543–571 MULTISCALE FINITE ELEMENTS FOR TRANSIENT ADVECTION-DIFFUSION EQUATIONS THROUGH ADVECTION-INDUCED COORDINATES \ KONRAD SIMON † AND J ¨ ORN BEHRENS † Abstract. Long simulation times in climate science typically require coarse grids due to compu- tational constraints. Nonetheless, unresolved subscale information significantly influences the prog- nostic variables and cannot be neglected for reliable long-term simulations. This is typically done via parametrizations, but their coupling to the coarse grid variables often involves simple heuristics. We explore a novel upscaling approach inspired by multiscale finite element methods. These methods are well established in porous media applications, where mostly stationary or quasi stationary situ- ations prevail. In advection-dominated problems arising in climate simulations, the approach needs to be adjusted. We do so by performing coordinate transforms that make the effect of transport milder in the vicinity of coarse element boundaries. The idea of our method is quite general, and we demonstrate it as a proof-of-concept on a one-dimensional passive advection-diffusion equation with oscillatory background velocity and diffusion. Key words. multiscale method, finite element method, advection-diffusion equation, subscale parametrization, upscaling AMS subject classifications. 34E13, 68U20, 65M60 DOI. 10.1137/18M117248X 1. Introduction. 1.1. Motivation and overview. Geophysical processes in our atmosphere and in the oceans involve many different spatial and temporal scales. Reliable climate sim- ulations therefore need to take into account the interaction of many scales since these are coupled and influence each other. Resolving all relevant scales and their interac- tions poses immense computational requirements. Even on modern high-performance computers, computational constraints force us to make compromises, i.e., to use a lim- ited grid resolution, which then often neglects certain scale interactions, or to model interactions in a heuristic manner. For example, long-term climate simulations, as done in the simulation of paleocli- mate, use grid resolutions of approximately 200 km or more. This, of course, ignores fine-scale processes that cannot be resolved by this resolution, although they signif- icantly influence prognostic variables on the coarse grid. Examples include (but are not limited to) moving ice-shields, land–sea boundaries, flow over rough orography, cloud physics, and precipitation. None of these processes is resolved by the grid, and current climate simulations cope with this by using so-called parametrizations. These can be seen as replacements or simplifications of subgrid processes or processes that are too complex to be taken into account on the prognostic (coarse) scale. Coupling * Received by the editors February 23, 2018; accepted for publication (in revised form) January 6, 2020; published electronically April 8, 2020. https://doi.org/10.1137/18M117248X Funding: This work was supported by the German Federal Ministry of Education and Research (BMBF) as Research for Sustainability initiative (FONA), www.fona.de, through Palmod project FKZ: 01LP1513A. † Department of Mathematics, Center for Earth System Research and Sustainability (CEN), Uni- versity of Hamburg, 20144 Hamburg, Germany (konrad.simon@uni-hamburg.de, joern.behrens@uni- hamburg.de). 543 Downloaded 05/08/20 to 134.100.237.161. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php