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

翻译：求和分部算子能在时间依赖微分方程中系统性地构造能量稳定且高阶精度的数值方法。近年来，现有SBP算子的核心思想在于多项式能够精确逼近解，因此SBP算子应对多项式保持精确性。然而对于某些问题，多项式并非最优逼近，其他逼近空间更为适用。我们近期提出这一问题的解决方案，发展了基于一般函数空间的一维SBP算子理论，并将其命名为函数空间SBP算子。本文将此理论拓展至多维情形，重点研究其存在性、与求积法则的关联、构造方法及模拟特性。关于多维函数空间SBP算子的更全面数值验证及其应用将在后续工作中呈现。与一维情况相似，我们证明基于多项式的多维SBP算子的大部分成熟结论可迁移至更广义的MFSBP算子。研究表明，SBP算子的应用范畴可显著超越现有方法体系，这既能提升数值解精度，又能增强方法的稳定性。