  Highly accurate simulations of problems including second derivatives on
complex geometries are of primary interest in academia and industry. Consider
for example the Navier-Stokes equations or wave propagation problems of
acoustic or elastic waves. Current finite difference discretization methods are
accurate and efficient on modern hardware, but they lack flexibility when it
comes to complex geometries. In this work I extend the continuous
summation-by-parts (SBP) framework to second derivatives and combine it with
spectral-type SBP operators on Gauss-Lobatto quadrature points to obtain a
highly efficient discretization (accurate with respect to runtime) of the
Laplacian on complex domains. The resulting Laplace operator is defined on a
grid without duplicated points on the interfaces, thus removing unnecessary
degrees of freedom in the scheme, and is proven to satisfy a discrete
equivalent to Green's first identity. Semi-discrete stability using the new
Laplace operator is proven for the acoustic wave equation in 2D. Furthermore,
the method can easily be coupled together with traditional finite difference
operators using glue-grid interpolation operators, resulting in a method with
great practical potential. Two numerical experiments are done on the acoustic
wave equation in 2D. First on a problem with an analytical solution,
demonstrating the accuracy and efficiency properties of the method. Finally, a
more realistic problem is solved, where a complex region of the domain is
discretized using the new method and coupled to the rest of the domain
discretized using a traditional finite difference method.
