Projections, Embeddings and Stability

Abstract: 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. The stability results are valid for general, nondiagonal summation-by-parts norms. Another key result is the extension of summation-by-parts operators to multi-domains by means of carefully crafted embedding operators. No extra numerical boundary conditions are required at the grid interfaces. The aforementioned pseudoinverse allows for a compact representation of these multi-block 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.