A posteriori error bounds for pseudo-parabolic equations using $C_0$ semigroups

Abstract: A class of pseudo-parabolic partial differential equations is considered. We derive a posteriori error bounds for approximations obtained by FEMs in space and a BDF formula in time. The analysis is based on the $C_0$ semigroup theory and an adaptation of the concept of elliptic reconstruction to pseudo-parabolic problems. The analysis is complemented with numerical experiments.

• The paper introduces a posteriori error bounds for pseudo-parabolic PDEs using C0 semigroup theory and an adapted elliptic reconstruction approach.
• It develops detailed error estimators in both H1 and L2 norms, demonstrating optimal convergence rates and robust efficiency indices.
• Numerical experiments validate the theoretical findings, providing practical tools for adaptive mesh refinement and reliable error control.

A Posteriori Error Bounds for Pseudo-Parabolic Equations via $C_0$ Semigroups

Problem Framework

The paper addresses a class of pseudo-parabolic PDEs formulated as $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$, with second-order elliptic, time-independent operators $\mathcal{L}$, $\mathcal{M}$ on $H^1_0(\Omega)$, and $F \in C([0,T]; H^{-1}(\Omega))$. The operator $\mathcal{M}$ is bounded, while $\mathcal{L}$ is bounded, coercive, and symmetric. The analysis focuses on a posteriori error estimation for numerical solutions generated via FEMs in space and BDF temporal discretization, distinguishing from prior work oriented towards a priori estimates.

Pseudo-parabolic equations arise in various applied and mathematical contexts, notably modeling time-dependent phenomena with memory effects and non-standard diffusion.

Analytical Tools and Theoretical Foundations

Uniqueness and regularity are established via Lax-Milgram, functional analytic reformulation, and $C_0$ semigroup theory. The operator $-\mathcal{L}^{-1}\mathcal{M}$ generates a uniformly continuous semigroup $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$0, which is used to represent solutions and derive stability estimates in the $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$1 norm. Precise norm estimates for $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$2 and its derivatives are provided, accommodating cases where the operator’s coercivity parameter $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$3 may be negative, an important generalization over classical parabolic theory.

A significant theoretical advancement is the adaptation of elliptic reconstruction to the pseudo-parabolic setting. Classical elliptic reconstruction, pioneered for parabolic equations, hinges on coercive elliptic operators, which do not directly apply to pseudo-parabolic equations due to the non-coercive structure of $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$4. This paper instead employs $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$5 alone for the reconstruction, enabling the transfer of elliptic a posteriori error estimators into this framework.

Additionally, a variant of the Green's function—based on the $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$6 semigroup—is constructed for $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$7 norm error analysis, facilitating sharper bounds thanks to higher regularity assumptions and functional representation of solution norms.

A Posteriori Error Estimators

The paper systematically develops error bounds for fully discrete schemes, both in $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$8 and $$K u(t) = \mathcal{L} \partial_t u(t) + \mathcal{M} u(t) = F(t)$$9 norms:

$\mathcal{L}$0 Norm Estimator:

A detailed decomposition exposes the main error components:

• Initialization error, propagated via the semigroup.
• Error originating from data approximation.
• Error incurred from time discretization artifacts (via BDF formulas).
• Elliptic error, measured by reconstruction difference and controlled via a posteriori estimators from the elliptic problem.
• Difference between FEM solution and elliptic reconstruction, including time-stepping effects.

The $\mathcal{L}$1 estimator combines these contributions with precisely calculated coefficients (involving the semigroup bounds), yielding strict computable upper bounds for the solution error at final time.

$\mathcal{L}$2 Norm Estimator:

Utilizing the Green's function-like representation, the $\mathcal{L}$3 norm of the error is expressed in terms of bilinear forms evaluated against the reconstructed solution and interpolated numerical data. Estimation uses functional analytic projection arguments and regularity results (for $\mathcal{L}$4), which are leveraged to obtain higher-order (in $\mathcal{L}$5) error bounds.

A standout aspect is the provision for higher convergence rates in the $\mathcal{L}$6 norm—explicitly one order faster than in $\mathcal{L}$7 for conforming FEMs—closely reflecting theoretical expectations.

Numerical Evidence

Numerical experiments substantiate theoretical claims. The $\mathcal{L}$8 estimator shows convergence rate $\mathcal{L}$9 (matching BDF-2) and efficiency indices between $\mathcal{M}$0 and $\mathcal{M}$1, demonstrating robust correlation between error and estimator. The dominant contributors are the data approximation and elliptic error components.

For the $\mathcal{M}$2 norm, the efficiency index is high—likely due to conservative estimates of unknown constants—but convergence order remains consistently two. The methodology for estimator computation leverages higher-order FEM for elliptic reconstruction and Simpson's rule for data integration, confirming practical adaption feasibility.

Implications and Future Directions

The paper establishes a rigorous and general approach for a posteriori error estimation in pseudo-parabolic equations with broad operator classes and full discretization. The adaptation of elliptic reconstruction to pseudo-parabolic contexts opens the way for a posteriori analysis of more complex evolutionary PDEs, including those with non-standard diffusive terms and coupled systems.

Practical implications include reliable error control for computational models involving pseudo-parabolic processes, allowing for mesh refinement and error-driven adaptive strategies in simulation software. The explicit bounds facilitate not only reliability but potential efficiency improvements in computational algorithms.

Theoretically, this work demonstrates the effective merger of semigroup theory, elliptic reconstruction, and functional analysis for error analysis. The approach may be extended to systems with variable coefficients, non-smooth domains, or more general time discretizations (e.g., higher-order BDF, discontinuous Galerkin). Investigation of optimal constants for projection and semigroup bounds, as well as extension to non-linear variants, are promising future directions.

Conclusion

This paper develops a comprehensive framework for a posteriori error bounds in pseudo-parabolic equations using $\mathcal{M}$3 semigroups and an adapted elliptic reconstruction. The estimators are computable, theoretically grounded, and substantiated through numerical experiments. These results provide significant tools for reliable numerical simulation and analysis of pseudo-parabolic PDEs, with broad application potential and promising avenues for further research in computational PDEs and numerical analysis (2606.20073).