  Many applications rely on solving time-dependent partial differential
equations (PDEs) that include second derivatives. Summation-by-parts (SBP)
operators are crucial for developing stable, high-order accurate numerical
methodologies for such problems. Conventionally, SBP operators are tailored to
the assumption that polynomials accurately approximate the solution, and SBP
operators should thus be exact for them. However, this assumption falls short
for a range of problems for which other approximation spaces are better suited.
We recently addressed this issue and developed a theory for first-derivative
SBP operators based on general function spaces, coined function-space SBP
(FSBP) operators. In this paper, we extend the innovation of FSBP operators to
accommodate second derivatives. The developed second-derivative FSBP operators
maintain the desired mimetic properties of existing polynomial SBP operators
while allowing for greater flexibility by being applicable to a broader range
of function spaces. We establish the existence of these operators and detail a
straightforward methodology for constructing them. By exploring various
function spaces, including trigonometric, exponential, and radial basis
functions, we illustrate the versatility of our approach. The work presented
here opens up possibilities for using second-derivative SBP operators based on
suitable function spaces, paving the way for a wide range of applications in
the future.
