A Posteriori Error Analysis of Finite Element Approximation to an Evolution Equation with a Positive-Type Memory Term

Sandhya Mel; Amiya Kumar Pani; Gujji Murali Mohan Reddy

Source-linked AI summary

A Posteriori Error Analysis of Finite Element Approximation to an Evolution Equation with a Positive-Type Memory Term

Sandhya Mel, Amiya Kumar Pani, Gujji Murali Mohan Reddy

arXiv:2610.37264v1Oct 27, 2026math.NAcs.NA

TL;DR

A posteriori estimates for positive-type memory equations have lacked fully discrete finite element bounds that account for both spatial and temporal errors. This paper combines elliptic reconstruction with finite element discretization and backward Euler quadrature, deriving robust estimates and demonstrating adaptive performance on an L-shaped benchmark.

Problem
Fully discrete finite element a posteriori estimates accounting for both spatial and temporal errors had not been derived for these memory equations, which also present regularity and storage challenges.
Source: §1 · p. 3
Method
The analysis combines elliptic reconstruction for spatial error control with finite element schemes and backward Euler time discretization using positive-type quadrature.
Source: §3 · p. 6, §4 · p. 9, §6 · p. 27
Results
The adaptive L-shaped-domain algorithm reduces error at a rate close to the inverse square root of the total number of unknowns.
Source: §1 · p. 4
Takeaways & Limitations
The paper establishes optimal-order a posteriori estimates for semidiscrete and fully discrete backward Euler approximations in the L∞(0,T;L2(Ω)) norm.
Source: §6 · p. 27
On the nonconvex L-shaped domain, the regularity assumption fails, so the reliability theorem does not apply directly and the experiment demonstrates adaptivity rather than bound reliability.
Source: §5.2.1 · p. 21

Abstract

from arXiv · show

This paper develops reliable a posteriori error estimates for finite element approximation of an evolution equation with a positive-type memory kernels on bounded convex polygonal or polyhedral domains, including both smooth and weakly singular kernels. For the weakly singular kernel $t^{\alpha-1}/\Gamma(\alpha),\; 0<\alpha<1,$ the model captures the transition from parabolic diffusion to wave propagation, recovering the corresponding limiting equations as $\alpha\to0^+$ and $\alpha\to1^-$, respectively, with the memory term governing the intermediate dynamics. The spatial discretization employs time-dependent finite element spaces, while the backward Euler method combined with numerical quadrature is used for time integration. A unified \emph{a posteriori} analysis is established for both semidiscrete and fully discrete approximations. Elliptic reconstruction yields optimal spatial error estimates, while a memory-lag estimate preserves the essential cancellation in the convolution term, yielding a first-order temporal estimator whose kernel dependence is characterized by its integral. The resulting estimators are robust, with constants independent of the final time, spatial mesh size, and time step. Their kernel dependence involves only computable quantities, avoiding exponential growth in time. Moreover, for the weakly singular kernel, the $\alpha$-dependent factors remain uniformly bounded for $\alpha\in(0,1).$ Numerical experiments on a convex domain validate the theoretical results, and a space-time adaptive algorithm is also demonstrated on an L-shaped domain featuring a corner singularity and two temporal layers.

1 Introduction

The introduction motivates a posteriori error analysis for positive-type memory equations and identifies the lack of fully discrete finite element estimates covering both spatial and temporal errors. It presents a unified analysis based on elliptic reconstruction and a memory-lag estimate, with robust bounds and numerical validation.

1 Introduction

The model covers smooth and weakly singular kernels; for K(t) = t^(α−1)/Γ(α), its dynamics transition from parabolic as α approaches 0 to wave-like as α approaches 1.Intermediate values of α describe dynamics in which the memory term reflects both diffusive and wave-like behavior.Source: §1 · pp. 1–2
Positive-type memory equations model phenomena including heat conduction in materials with memory and compressible viscoelastic media, while adding regularity and storage challenges.The memory term distinguishes these problems from purely parabolic or hyperbolic equations.Source: §1 · pp. 2–3
Fully discrete finite element estimates accounting for both spatial and temporal errors were not previously available for the target equation, motivating this work.Earlier time-discretization estimates were restricted to discrete time levels and included a logarithmic factor depending on final time and the smallest step.Source: §1 · p. 3
The backward Euler scheme with positive-type quadrature supports fully computable bounds on time-varying finite element meshes, including an estimator for mesh changes in the discrete history.The history mesh-change estimator vanishes when the mesh is unchanged.Source: §1 · p. 3
Elliptic reconstruction and a memory-lag estimate combine spatial error control with treatment of temporal nonlocality in the fully discrete analysis.The memory-lag estimate retains cancellation between memory at the current time and at the discrete time level.Source: §1 · pp. 3–4
The bounds avoid Gronwall-based exponential growth, have constants independent of final time, mesh size, and time step, and use computable kernel-dependent quantities.For the weakly singular kernel, the α-dependent estimator factors remain uniformly bounded for α in (0,1).Source: §1 · p. 4
The adaptive L-shaped-domain experiment refines near the corner and temporal layers and reduces error at a rate close to the inverse square root of the total unknowns.The benchmark combines a corner singularity with two temporal layers.Source: §1 · p. 4

2 Notation and preliminaries

This section establishes the function-space and operator setting for the variational problem. It assumes a convex polygonal or polyhedral domain, a coercive elliptic bilinear form, and an L2 elliptic regularity property.

2 Notation and preliminaries

The analysis uses Lp and Sobolev spaces, including the Bochner spaces L2(0,T;L2(Ω)) and L∞(0,T;L2(Ω)).The variational solution is sought in L∞(0,T;H1_0(Ω)) with time derivative in L2(0,T;L2(Ω)).Source: §2 · pp. 4–5
The analysis assumes that a(·,·) is continuous and coercive on H1_0(Ω); under the stated condition on coefficient matrix {aij(x)}1≤i,j≤d, the bilinear form corresponding to A has these properties.Source: §2 · p. 5
An elliptic H2-regularity property is assumed, with its constant independent of the meshes, time steps, and memory kernel.This regularity assumption is stated for weak solutions with L2 data.Source: §2 · p. 5

3 Semidiscrete scheme

The semidiscrete section analyzes a spatial Galerkin finite element approximation using elliptic reconstruction to separate reconstruction and temporal errors. It derives an L∞(0,T;L2(Ω)) a posteriori bound without Gronwall’s lemma.

3 Semidiscrete scheme

The spatial approximation uses continuous piecewise linear finite elements, with the analysis also extending to any polynomial degree.The meshes are conforming and shape-regular.Source: §3 · pp. 5–6
The spatial residual construction uses the elementwise regular part of the elliptic operator and jumps of the flux across internal mesh sides.The mesh-side jump is defined for A∇v across an internal edge or face.Source: §3 · p. 6
Elliptic reconstruction splits the main error into a reconstruction error and a temporal error, which are estimated separately before being combined.The decomposition is e = ρ − ε, with ε the reconstruction error and ρ the temporal error.Source: §3 · p. 6
The reconstruction error and its time derivative receive separate bounds, which feed into the temporal error estimate and the final theorem.The temporal estimate uses the positive definiteness of the kernel in its energy argument.Source: §3 · pp. 7–8
The semidiscrete theorem bounds the error in L∞(0,T;L2(Ω)) for every time from 0 to T, without invoking Gronwall’s lemma.Its constants are independent of final time, mesh parameters, and the kernel.Source: §3 · p. 8

4 Analysis for the fully discrete scheme

This section formulates the fully discrete backward Euler scheme with positive-type quadrature on time-dependent meshes and develops its a posteriori error analysis, decomposing the error into reconstruction and temporal components and bounding spatial, temporal, quadrature, and memory-related contributions. It establishes a space-time L∞(L²)-norm error bound for U, with reliability constants for weakly singular kernels independent of α, mesh sizes, and time steps, and extends the analysis to nonconvex domains using regularity-dependent weights.

4 Analysis for the fully discrete scheme

The fully discrete scheme uses backward Euler time stepping and a positive-type quadrature rule for the memory integral on a sequence of time-dependent finite element meshes.Consecutive meshes are assumed compatible refinements of a common coarsening, with a uniformly bounded mesh-ratio condition.Source: §4 · pp. 8–9
A piecewise-linear-in-time approximation and elliptic reconstruction are used to split the total error into temporal and reconstruction errors.The reconstruction is formed by interpolating the elliptic reconstructions of consecutive backward Euler solutions.Source: §4 · pp. 10–11
The analysis defines separate estimators for elliptic reconstruction, spatial discretization, time discretization, quadrature, memory consistency, and mesh changes.The memory-consistency estimator accounts for memory lag and history mesh changes.Source: §4 · pp. 10–11
The spatial error bound holds under the mesh assumptions, with estimator constants independent of individual mesh sizes, time steps, and the memory kernel.The constant for the additional-side trace estimate may depend on the fixed mesh-ratio bound L.Source: §4 · p. 13
The temporal error analysis bounds the time-discretization contribution, while the quadrature error is estimated separately for smooth and weakly singular kernels.The quadrature estimator uses a computable kernel quantity.Source: §4 · pp. 11, 13, 14, 15
The memory-consistency analysis separates lag from history mesh change; the lag estimate preserves cancellation by comparing discrete histories at nearby time arguments.It also accounts for memory accumulated over the short interval between t and the next time node.Source: §4 · p. 16
The analysis establishes auxiliary error bounds for the mesh-change error and ρ, which are used in proving the fully discrete estimate.Lemma 4.8 gives the mesh-change estimate, and Lemma 4.9 provides an estimate for ρ.Source: §4 · p. 17
Theorem 4.1 gives a space-time L∞(L2)-norm a posteriori error bound for the fully discrete approximation U at each time level.The theorem assumes u satisfies (1.1) and U is given by (4.7).Source: §4 · p. 18
For weakly singular kernels, reliability constants are independent of α, individual mesh sizes, and time steps, while kernel-dependent estimator factors remain uniformly bounded for α ∈ (0,1).The constants may depend on elliptic regularity, mesh shape-regularity, and the fixed relative-refinement bound L; for a fixed time grid, the factors and quadrature weights have finite endpoint limits.Source: §4 · p. 18
On a nonconvex polygon with largest interior angle ω, the proof extends using regularity 0 < s < π/ω and replacing the convex-domain weights with h^(1+s) and h^(1/2+s).The L-shaped experiment in Section 5.2 uses the original weights.Source: §4 · p. 18

5 Numerical experiments

The experiments assess estimator convergence, space–time adaptivity, and dependence on the fractional parameter. Their implementation uses conforming P1 finite elements and backward Euler time stepping, with estimator constants set to one.

5 Numerical experiments

The experiments examine estimator convergence, space–time adaptivity, and dependence on the fractional parameter α.Computations use FreeFEM and MATLAB.Source: §5 · p. 19
All computations use conforming P1 finite elements and backward Euler time stepping.Source: §5 · p. 19
Because implementation sets the unevaluated reconstruction and interpolation constants to one, reported values assess performance rather than certified upper bounds.Source: §5 · p. 19

5.1 Convergence on uniform meshes

The uniform-mesh experiments couple spatial and temporal refinement, test smooth and weakly singular kernels, and show approximately first-order decay in element count for errors and estimator contributions.

5.1 Convergence on uniform meshes

Uniform spatial meshes and time steps τ ∼ h2 balance expected spatial and temporal errors, so these tests assess combined convergence rather than isolate either order.Separate spatial and temporal studies would be needed to identify the two orders.Source: §5.1 · p. 19

5.1.1 Smooth kernel

The smooth-kernel test extends the benchmark from to two dimensions and uses K(t − s) = e−2(t−s).Source: §5.1.1 · p. 19

5.1.2 Weakly singular kernel

The weakly singular test follows, uses K(t − s) = (t−s)^(α−1)/Γ(α) with α = 1/2, and chooses the source to give an exact solution.Source: §5.1.2 · p. 19

5.1.3 Error and estimator convergence

Approximately first-order decay with element count appears for the error and estimator contributions in both kernel examples, consistent with O(h2 + τ) under τ ∼ h2.The different error magnitudes cannot be attributed solely to the kernel because the examples use different exact solutions and source terms.Source: §5.1.3 · p. 20, Figure 1 · p. 20

5.2 Space–time adaptivity on an L-shaped domain

The L-shaped benchmark tests space–time adaptivity on a nonconvex domain with a corner singularity and two temporal layers, outside the direct scope of Theorem 4.1. Adaptive runs show similar error-versus-work slopes to uniform runs but a lower error level in the reported comparison.

5.2.1 Benchmark problem

The L-shaped domain introduces a re-entrant corner singularity and two temporal layers to test space–time adaptivity under reduced regularity.The domain is nonconvex, so the regularity assumption fails and Theorem 4.1 does not apply directly; the experiment investigates adaptivity rather than verifying the reliability bound there.Source: §5.2.1 · pp. 20–21

5.2.2 Adaptive indicators and algorithm

Spatial indicators use reconstruction and data-oscillation contributions, while the temporal indicator combines data variation, a memory-weighted discrete time derivative, and the kernel-primitive increment.The algorithm selects meshes and time steps using these local indicators; trial steps failing either spatial criterion are recomputed after refinement.Source: §5.2.2 · pp. 21–22

5.2.3 Implementation and reported quantities

Table 1 reports adaptive and uniform runs using work W, sampled error, estimator, time-step counts, final degrees of freedom, and experimental convergence orders.The sampled error is evaluated at time levels and interval midpoints and approximates, but need not equal, the continuous-in-time L∞(0,T;L2(Ω)) error.Source: §5.2.3 · p. 22, Table 1 · p. 23

5.2.4 Adaptive meshes, time steps and convergence

Time steps are graded at both temporal layers, and each halving of tolerances increases the step count by about 1.4; the mesh refines near the corner and cutoff-data ring.The layer at t = 1/2 is detected through the temporal indicator; refinement is strongest near the re-entrant corner and in 0.05 < r < 0.45.Source: §5.2.4 · p. 23
A0 reaches error 1.01·10−3 with W = 2.59·105, versus U32’s 2.46·10−3 with W = 1.19·105; this is 2.4 times smaller error for about 2.2 times the work.The adaptive sequence does not improve the convergence rate with respect to W; across A0–A2, error and estimator slopes are approximately −0.45 and −0.50.Source: §5.2.4 · pp. 23, 25, Figure 4 · p. 25
The memory-lag term contributes 73–74% of the adaptive estimator, making it conservative because it retains large discrete derivatives from the initial layer.Effectivity indices depend on constants set to one and should therefore be interpreted with care.Source: §5.2.4 · p. 25

5.3 Dependence on α

Across seven tested values of α, including values near both limits, the experiments compare fixed adaptive and uniform refinements. Estimator ranges and refinement reductions remain similar across α.

5.3 Dependence on α

Across all tested α values, A0→A2 reduces the estimator by 0.48–0.52 and error by 0.46–0.59; U16→U32 reduces them by 0.26–0.27 and 0.31–0.32.The tested reductions are similar across α for both adaptive and uniform refinement.Source: Figure 5 · p. 26, §5.3 · p. 26
The passage reports 1.88 for A0 and estimator values between 0.80 and 0.96 for A2, also at α=0.001 and α=0.999.Estimates at α=0.001 and 0.01 (1.882 and 1.844), and at α=0.99 and 0.999 (1.785 and 1.778), almost coincide, consistent with finite limits of the weights.Source: §5.3 · p. 26
A0’s error ranges from 7.61·10−4 to 1.36·10−3, while its time-step count ranges from 99 to 126.The step count decreases from 126 to 99 as α increases from 0.1 to 0.999.Source: §5.3 · p. 27

6 Conclusion

The paper establishes optimal-order a posteriori estimates for positive-type memory evolution equations, with robust bounds and an adaptive algorithm demonstrated on a singular space-time solution.

6 Conclusion

Optimal-order a posteriori estimates cover smooth and weakly singular kernels for semidiscrete and fully discrete backward Euler schemes in L∞(0,T;L2(Ω)).The upper-bound constants are independent of final time, mesh size, time steps, and kernel; weakly singular-kernel factors remain uniformly bounded for α∈(0,1).Source: §6 · p. 27
Elliptic reconstruction extends to positive-type memory equations, while the fully discrete analysis uses memory-lag cancellation to obtain a first-order temporal estimator depending on the kernel through K1.The bounds avoid exponential growth in time.Source: §6 · p. 27
For a solution with a corner singularity and two temporal layers, estimator-driven adaptation refines the corner and grades time steps at both layers.Source: §6 · p. 27