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Numerical Treatment of Non-local Integral Operators in the Framework of Evolutionary Equations

Sebastian Franz, Sascha Trostorff

TL;DR

This work extends the abstract evolutionary equation framework to include a non-local integral operator $T_K$ with kernel $K$, proving well-posedness and causality under a kernel-bound condition in a weighted Hilbert space $L^{2,\rho}(\mathbb{R};H)$. It then develops a space-time Galerkin discretisation (conforming finite elements in space and discontinuous Galerkin in time) and provides a rigorous convergence analysis, showing $\|U-U^{\tau,h}\|_{\rho} \le C(h^k + T^{1/2}\tau^{q+1})$ under suitable regularity and kernel assumptions. Numerical experiments with smooth and weakly singular kernels corroborate the theoretical rates and illustrate practical considerations for history-term computations and adaptive quadrature. The results lay a solid foundation for numerically solving Volterra/Fredholm-type integro-differential problems within the Picard framework, with potential extensions to time-derivative kernels and more general non-local operators.

Abstract

Using the theory of evolutionary equations, we consider abstract differential equations including non-local integral operators. After providing a condition for the well-posedness of the addressed equation we consider a numerical method of approximating its solution. We provide convergence proofs under conditions on the kernel of the integral operator and the solution and finish the paper with some simulation results.

Numerical Treatment of Non-local Integral Operators in the Framework of Evolutionary Equations

TL;DR

This work extends the abstract evolutionary equation framework to include a non-local integral operator with kernel , proving well-posedness and causality under a kernel-bound condition in a weighted Hilbert space . It then develops a space-time Galerkin discretisation (conforming finite elements in space and discontinuous Galerkin in time) and provides a rigorous convergence analysis, showing under suitable regularity and kernel assumptions. Numerical experiments with smooth and weakly singular kernels corroborate the theoretical rates and illustrate practical considerations for history-term computations and adaptive quadrature. The results lay a solid foundation for numerically solving Volterra/Fredholm-type integro-differential problems within the Picard framework, with potential extensions to time-derivative kernels and more general non-local operators.

Abstract

Using the theory of evolutionary equations, we consider abstract differential equations including non-local integral operators. After providing a condition for the well-posedness of the addressed equation we consider a numerical method of approximating its solution. We provide convergence proofs under conditions on the kernel of the integral operator and the solution and finish the paper with some simulation results.
Paper Structure (9 sections, 7 theorems, 81 equations, 1 figure, 3 tables)

This paper contains 9 sections, 7 theorems, 81 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Numerical method
  4. Numerical Analysis
  5. Error equation and coercivity
  6. Convergence analysis
  7. On the implementation
  8. Numerical examples
  9. Conclusion

Key Result

Theorem 1.1

Let $A:\operatorname{dom}(A)\subseteq H\to H$ be skew-selfadjoint, $M_0,M_1$ bounded operators on $H$ such that $M_0$ is selfadjoint. Moreover, assume that there exists $\rho_0\in \mathbb{R}$ and $c>0$ such that Then for each $\rho\geq \rho_0$ the problem eq:evo_eq is well-posed in the sense, that for each $F\in L^{2,\rho}(\mathbb{R};H)$ there exists a unique $U\in L^{2,\rho}(\mathbb{R};H)$ satis

Figures (1)

  • Figure 1: Solution $u$ (left) and $v$ (right) of Example 3

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 7 more