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Projections, Embeddings and Stability

Pelle Olsson

TL;DR

The paper develops an operator-centric SBP framework for linear PDEs by leveraging projections and Moore–Penrose pseudoinverses to represent boundary conditions. It introduces a simplified semidiscrete form that avoids time derivatives of boundary data and constructs stable, multi-block operators via embedding, enabling interface handling without extra boundary conditions. The approach extends to multi-dimensional problems including Maxwell’s equations, with energy-based stability guarantees and numerical validation. Overall, the work provides a rigorous, algebraic pathway to boundary and interface treatment in SBP discretizations with strong stability properties across complex multi-domain and multi-block settings.

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.

Projections, Embeddings and Stability

TL;DR

The paper develops an operator-centric SBP framework for linear PDEs by leveraging projections and Moore–Penrose pseudoinverses to represent boundary conditions. It introduces a simplified semidiscrete form that avoids time derivatives of boundary data and constructs stable, multi-block operators via embedding, enabling interface handling without extra boundary conditions. The approach extends to multi-dimensional problems including Maxwell’s equations, with energy-based stability guarantees and numerical validation. Overall, the work provides a rigorous, algebraic pathway to boundary and interface treatment in SBP discretizations with strong stability properties across complex multi-domain and multi-block settings.

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.
Paper Structure (58 sections, 55 theorems, 711 equations, 7 figures, 1 table)

This paper contains 58 sections, 55 theorems, 711 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Projections
  3. Adjoint operators
  4. Self-adjoint operators
  5. Least squares and the pseudoinverse
  6. Difference operators and summation by parts
  7. The solution state space $V$
  8. Initial-boundary value problems
  9. The boundary state space $V_\Gamma$
  10. The semidiscrete equations
  11. The simplified semidiscrete form
  12. Consistency of the semidiscrete approximation
  13. Boundary conditions and the pseudoinverse
  14. The simplified projection $P$
  15. Characteristic boundary conditions
  16. ...and 43 more sections

Key Result

Theorem 1

Let $x \in V$ be an inner product space (eq:scal) and let ${\cal L}$ be a linear manifold in $V$. There is a unique vector $\hat{x}\in{\cal L}$ such that

Figures (7)

  • Figure 1: Two uniform grids with states $u^{(1)}$ and $u^{(1)}$
  • Figure 2: Unit square with mesh size $h_1$ and $h_2$
  • Figure 3: Two blocks, case 1
  • Figure 4: Two blocks, case 2
  • Figure 5: Four blocks
  • ...and 2 more figures

Theorems & Definitions (101)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Definition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Definition 9
  • Proposition 10
  • ...and 91 more