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Sharp Error Estimates for a Fully Discrete Finite Element Method for Semilinear SPDEs with Multiplicative Noise and Nonsmooth Initial Data
Jitendra Nath Naik, Lok Pati Tripathi
TL;DR
For multiplicative-noise SPDEs, limited temporal regularity prevents attaining the spatial regularity available with additive noise under standard assumptions. The paper uses a fractional-space framework for its analysis. It establishes estimates for fixed positive times while allowing noise and initial-data regularity to be prescribed independently.
Problem
Under standard Lipschitz and linear-growth assumptions, multiplicative-noise SPDEs do not attain the H2 spatial regularity possible with additive noise, and their limited temporal regularity complicates numerical convergence analysis.
Source: §1 · pp. 2–3Method
The analysis recasts the equation in a Hilbert space and uses fractional spaces with a noise regularity parameter β.
Source: §1 · p. 2Results
For each fixed t > 0 and ε ∈ (0, β), the framework permits initial-data regularity μ and noise regularity β to be prescribed independently, under weaker coefficient assumptions than in.
Source: §1 · p. 4Takeaways & Limitations
Numerical experiments verify the predicted spatial and temporal convergence rates.
Source: §1 · p. 4The stated coefficient setting requires sufficiently smooth and bounded coefficients, with the principal coefficient symmetric and uniformly positive definite.
Source: §1 · p. 2
Abstract
from arXiv · showhide
This article establishes sharp strong error estimates for the fully discrete approximation of semilinear parabolic stochastic partial differential equations (SPDEs) driven by multiplicative noise and subject to nonsmooth initial data. The spatial discretization is based on a standard finite element method, coupled with the linearly implicit Euler scheme in time. By mapping the diffusion operator into negative fractional spaces, our framework accommodates both trace-class and space-time white noise. For nonsmooth initial data, by decoupling the noise regularity parameter $\beta \in (0,2)$ from the initial data regularity parameter $\mu \in (0,2]$, we derive sharp regularity estimates that isolate the exact loss of initial regularity into an integrable temporal singularity. Furthermore, we establish sharp strong convergence rates of $O(h^{\beta-\varepsilon} + k^{\frac{1}{2}\min\{\beta-\varepsilon, 1\}})$ for $\varepsilon>0$ away from $t = 0$. Finally, we consider physically relevant stochastic models, such as the modified Langmuir fractional surface coverage model and the parabolic Anderson model, in our numerical experiments to confirm the theoretical convergence rates.
1 Introduction
The introduction identifies regularity barriers in multiplicative-noise SPDEs and the added difficulty of obtaining sharp numerical convergence estimates with nonsmooth initial data. It presents a framework that separates noise and initial-data regularity, establishes sharp estimates for a fully discrete finite element method, and tests predicted rates numerically.
1 Introduction
- Multiplicative noise creates a spatial regularity barrier: under standard Lipschitz and linear growth assumptions, solutions do not attain H^2 regularity, complicating convergence analysis.The stated barrier is linked to solution time Hölder continuity with exponent below 1/2 and a non-integrable semigroup singularity.Source: §1 · pp. 2–3
- Earlier strong convergence results cited for multiplicative noise were restricted to trace-class noise, while sharp analysis for nonsmooth initial data had been achieved only for additive noise.The introduction identifies the combination of nonsmooth initial data and multiplicative space-time white noise as unaddressed.Source: §1 · p. 3
- Mapping the diffusion coefficient into HS(U0, ˙H^(β−1)) allows the framework to accommodate space-time white noise as well as trace-class noise.The negative-order space introduces decaying spectral weights that keep Hilbert–Schmidt norms finite for Q = I.Source: §1 · pp. 2–3
- The article establishes sharp spatial and temporal regularity estimates for the mild solution and presents numerical experiments to verify the predicted convergence rates.The experiments include the modified Langmuir fractional surface coverage model and the parabolic Anderson model.Source: §1 · pp. 3–4
- The fully discrete finite element and linearly implicit Euler approximation has a strong convergence estimate for each fixed t > 0, with noise regularity β and initial-data regularity μ prescribed independently.The estimate holds for ε ∈ (0, β) and under weaker assumptions on F and G than those imposed in.Source: §1 · p. 4
2 Setting and assumptions
The setting places the drift and diffusion in negative-order spaces and imposes assumptions to obtain a unique mild solution and regularity bounds. These choices also relax restrictions on the drift and allow diffusion coefficients compatible with space-time white noise.
2 Setting and assumptions
- The drift may take values in ˙H−1, relaxing standard codomain restrictions and allowing non-self-adjoint operators.The cited standard restrictions place the drift in H or ˙H^(−1+r) for r ≥ 0.Source: §2 · p. 5
- The diffusion is formulated in HS(U0, ˙H^(β−1)), rather than the standard HS(U0, H) space, enabling the framework to accommodate multiplicative space-time white noise.The noise regularity parameter β lies in (0, 2).Source: §2 · pp. 4–5
- Under Assumptions 2.1–2.3, problem (1.4) admits a unique mild solution X; furthermore, for p ∈ [2, ∞), the theorem states that additional properties hold.Source: §2 · p. 5
- The temporal regularity exponent can be taken below 1/2 min(1, μ, β), tying the bound to both initial-data and noise regularity.The proof derives this exponent using the initial data condition and the framework in [25].Source: §2 · p. 5
3 Fully discrete approximation
The fully discrete method projects the problem onto finite element spaces and applies a linearly implicit Euler time step. The section states a strong error estimate under conditions coupling the initial-data and noise regularities.
3 Fully discrete approximation
- The spatial discretization projects the SPDE onto continuous, piecewise linear finite element spaces V_h using a generalized orthogonal projector from ˙H−1.The spaces are finite-dimensional subspaces of ˙H1 on a regular triangulation.Source: §3 · p. 5
- A linearly implicit Euler step advances the semidiscrete problem on a uniform time grid using the resolvent S_(h,k) = (I + kA_h)^−1.The increments are ΔW^n = W(t_n) − W(t_(n−1)).Source: §3 · p. 6
- Expanding the discrete recursion yields a variation-of-constants representation of the fully discrete solution.The representation is stated for X_h^n at steps n = 1, …, N.Source: §3 · p. 6
- Theorem 3.1 gives a strong error estimate comparing the mild solution X(t_n) with the fully discrete approximation X_h^n, when μ > β − 1.It defines ν = min{β, μ} and applies for arbitrarily small ε ∈ (0, β), with a constant independent of h, k, and n.Source: §3 · p. 6
4 Regularity results
Theorem 4.1 establishes weighted spatial and temporal regularity estimates for the mild solution under conditions linking the noise regularity β and initial-data regularity μ. Its proof treats β above 1 separately, using a bootstrap argument.
4 Regularity results
- Under Assumptions 2.1–2.3, Theorem 4.1 gives spatial and temporal regularity estimates when β ∈ (0, 2), μ ∈ (0, 2], and μ > β − 1.The theorem defines ν as min{β, μ}.Source: §4 · p. 7
- For β ∈ (1, 2), the proof bootstraps the spatial estimate at order β − 1 and uses the embedding into the required fractional space.The condition μ > β − 1 ensures the parameter at the bootstrap order equals β − 1.Source: §4 · p. 8
- The temporal regularity proof sets r = max{β − 1, 0}, decomposes the time increment, and combines its component bounds to establish estimate (4.2).The decomposition covers the drift and diffusion terms as well as the first term of the increment.Source: §4 · pp. 9–10
5 Proof of Theorem 3.1
The proof builds the fully discrete strong error estimate from deterministic approximation bounds and estimates for the drift and diffusion contributions. It combines these bounds and applies a discrete Grönwall inequality to obtain the final estimate.
5 Proof of Theorem 3.1
- For the deterministic component, the proof compares the exact solution u(t) = S(t)v with its fully discrete approximation using the linearly implicit Euler method.The auxiliary estimates cover the fully discrete approximation for t_n > 0 and h, k ∈ (0, 1].Source: §5 · p. 10
- The drift and diffusion error terms are each split into three components, with bounds using the regularity estimates and deterministic approximation lemmas.The proof also invokes a generalized discrete fractional Grönwall lemma to combine the estimates.Source: §5 · pp. 10, 11, 13
- The proof combines the initial-data, drift, and diffusion error bounds to control the total strong error between X(t_n) and X_h^n.The final estimate follows after taking a square root and applying the discrete Grönwall inequality.Source: §5 · p. 15
6 Numerical experiments
The experiments test the method across noise regularities and smooth and nonsmooth initial data using finite elements in space and linearly implicit Euler in time. For nonsmooth starts, time-weighted errors recover the stated sharp rates, while the standard errors do not; with smooth starts, both metrics show identical convergence behavior.
6 Numerical experiments
- The simulations cover trace-class noise at β=1.0 and 1.99 and space-time white noise at β=0.4995, with Assumption 2.2 satisfied in all cases.The white-noise case uses Q=I; the trace-class cases satisfy the stated constraint.Source: §6 · pp. 16–17
- The SPDEs use continuous piecewise-linear finite elements on uniform spatial grids and a linearly implicit Euler method in time.The reference noise realization is shared across coarsened spatial grids to isolate spatial discretization error.Source: §6 · p. 17
- For nonsmooth starts, the discrete time weight tn absorbs the initial singularity and allows the weighted metric to test the theoretical sharp rate.The experiments compare standard and time-weighted errors; smooth starts have no transient singularity and show identical behavior under both metrics.Source: §6 · p. 17
6.1 Smooth initial data
- With smooth initial data u0(x)=sin(πx), spatial and temporal rates match O(h^(β−ε) + k^(1/2 min{β−ε,1})), and standard and time-weighted metrics behave identically.The reported rate is for ε>0.Source: §6.1 · p. 18, Figure 1 · p. 19
6.2 Nonsmooth initial data
- For nonsmooth starts in the modified Langmuir and parabolic Anderson models, time-weighted errors recover O(h^(β−ε) + k^(1/2 min{β−ε,1})) while standard errors do not.The reported recovery holds across tested noise regularities.Source: §6.2 · pp. 19–20, Figure 2 · p. 20, Figure 3 · p. 21