Description
Mimetic finite-differences (MFD) are discrete analogs of the differential operators used to describe continuum problems, with successful implementations in the fields of fluid and solid mechanics. Recent developments have focused on employing MFD to solve challenging problems by targeting partial differential equations (PDEs) with rough coefficients, jump discontinuities, and highly nonlinear problems. However, the use of MFD on complex geometries has not been studied in detail. The curvilinear method is a grid discretization technique commonly applied in conjunction with finite-difference or related schemes to deal with irregular geometries. This method consists of mapping the complex physical region into a more straightforward computational domain. Another approach employed to solve problems on non-trivial geometries is overlapping grids. This technique focuses on dividing the problem domain into subregions using structured curvilinear grids that overlap. Finally, overlapping grids are regularly used on problems that involve the simulation of complex and moving geometry. In essence, this research focuses on studying the viability of employing MFD schemes on non trivial problems. With non-trivial, we refer to PDEs defined with rough coefficients, irregular domains, nonlinearities, or a combination of these categories. Rough coefficients are modeled by way of our novel and efficient implementation of the mimetic flux operator. Finally, we incorporate the curvilinear mapping transformation and the overlapping grids method into the mimetic framework to deal with irregular domains. We present numerical results to demonstrate the effectiveness of our schemes.
