Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP was that polynomials can accurately approximate solution, should thus be exact them. However, do not provide best approximation some problems, with other spaces being more appropriate. We recently addressed this issue developed a theory one-dimensional based on general function spaces, coined function-space (FSBP) operators. In paper, we extend of FSBP multiple dimensions. focus their existence, connection quadratures, construction, mimetic properties. A exhaustive demonstration multi-dimensional (MFSBP) application will provided in future works. Similar case, demonstrate most established results polynomial-based (MSBP) carry over class MFSBP Our findings imply concept applied significantly larger than is currently done. This increase accuracy solutions and/or stability methods.