Embedded boundary conditions in spectral methods: A rectangular matrix approach

Boundary conditions play a crucial role in determining well-posed solutions to differential equations. This paper introduces an innovative spectral method framework in Hilbert spaces that systematically embeds boundary conditions directly into the differential operator structure. Through the construction of modified operators ℒ δ (where δ denotes the highest derivative order), our approach streamlines numerical solutions while maintaining spectral accuracy, particularly benefiting high-order equations ( δ > 2 ). The methodology demonstrates remarkable versatility across a range of applications. It effectively handles linear and nonlinear differential equations, systems with constant or variable coefficients in R d ( d ∈ N ), as well as eigenvalue problems and variational formulations, supporting both nodal and modal spectral representations. By leveraging the intrinsic rectangular structure of operational matrices, our technique eliminates conventional constraint enforcement strategies like row deletion or interpolation. This fundamental rethinking of boundary condition implementation simplifies computation while potentially improving numerical stability and accuracy. Numerical experiments using Jacobi polynomials (with special emphasis on Legendre/Chebyshev bases) and Fourier basis functions validate the method’s effectiveness. We further develop a novel a posteriori error estimation framework that evaluates solution consistency without requiring prior knowledge of analytic solutions. The results establish our approach as a robust, unified framework for high-order differential equations with complex boundary conditions, offering advantages in implementation simplicity and computational efficiency.