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（SBP）算子使我们能够系统地发展针对时间依赖微分方程的能量稳定且高精度数值方法。直到最近，现有SBP算子的主要思想基于多项式能精确逼近解，因此SBP算子应对多项式保持精确。然而，对于某些问题，多项式并不能提供最佳逼近，其他逼近空间更为合适。我们近期解决了这一问题，并基于一般函数空间发展了一维SBP算子理论，命名为函数空间SBP（FSBP）算子。本文中，我们将FSBP算子理论推广至多维情形。重点研究其存在性、与求积规则的关联、构造及仿射性质。关于多维FSBP（MFSBP）算子的更详尽数值演示及其应用将在未来工作中给出。与一维情况类似，我们证明大多数基于多项式建立的多维SBP（MSBP）算子的结论可推广至更一般的MFSBP算子类。我们的发现表明，SBP算子的概念可应用于比当前实践更广泛的数值方法类别，这有助于提高数值解的精度和/或为方法提供稳定性。