Numerical methods for parabolic multiscale problems

• Applicant: Dr. Roland Maier
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 545165789

Many physical phenomena, such as diffusion processes in highly heterogeneous materials, are characterized by effects on multiple scales. These phenomena are typically modeled by partial differential equations involving highly oscillatory coefficients. To avoid expensive and unfeasible global computations on very fine discretization scales that resolve these oscillations, appropriate multiscale methods can be considered. These techniques operate on coarse scales and take fine quantities only locally into account. The treatment of multiscale problems for spatially oscillating coefficients is well-understood. The construction of higher-order methods under minimal regularity assumptions, however, is not straight-forward for time-dependent equations. Moreover, classical multiscale methods are not efficient or even fail to compute reliable approximations if rapid variations in time are present as well. This project is devoted to the design, implementation, and analysis of efficient multiscale methods for linear parabolic equations. The focus lies on the construction of higher-order methods as well as methods for highly oscillatory coefficients with respect to both space and time. The methods are constructed based on three main principles: locality, parallelism, and coarse-scale communication. Unfeasible global computations on microscopic scales will be replaced by the solution of locally defined time-dependent sub-problems. These problems are defined independently of each other, which allows for parallel computations, and their solutions are only coupled on coarse scales in space and time. The developed methods will be theoretically analyzed and numerically validated. Further, the incorporation of deep learning strategies to enhance the numerical methods, in particular in multi-query scenarios, will be studied. An important aspect will be reliable error estimators for the learning-enhanced methods, which allow to evaluate the discrete solutions and to adapt the construction if necessary.

• DFG Programme: Research Grants