Projection based summation-by-parts methods, embeddings and the pseudoinverse

In the present work, we demonstrate how the pseudoinverse concept from linear algebra can be used to represent and analyze the boundary conditions of linear systems of partial differential equations. This approach has theoretical and practical implications; the theory applies even if the boundary operator is rank deficient, or near rank deficient. If desired, the pseudoinverse can be implemented directly using standard tools like Matlab. We also introduce a new and simplified version of the semidiscrete approximation of the linear PDE system, which completely avoids taking the time derivative of the boundary data, cf. [1]. The 2D stability results of the projection method in [2] are extended to nondiagonal summation-by-parts norms, which introduce boundary terms that require special attention in case of the projection method (equivalence of diagonal and nondiagonal boundary norms), see [3] for details. Another key result is the extension of summation-by-parts operators to multidomains by means of carefully crafted embedding operators. No extra numerical boundary conditions are required at the grid interfaces. The pseudoinverse allows for a compact representation of these multiblock operators, which preserves all relevant properties of the single-block operators. The embedding operators can be constructed for multiple space dimensions. Numerical results for the two-dimensional Maxwell’s equations are presented, and they show very good agreement with theory.