Convergence of finite element approximations for the one-dimensional stochastic Burgers equation with additive trace-class noise

1 Introduction

• Finite-element analyses had not systematically treated both strong and weak errors, while explicit strong rates remained largely L2-focused and lacked space-time maximum-norm results.Earlier explicit rates for fully discrete schemes used spectral Galerkin space discretization with tamed exponential Euler time stepping [38] [39].Source: §1 · p. 2
• P2 semi-discretization attains h^(1−α) regularity-optimal strong rates in Lp([0,T]×Ω;H_D^(α,q)) and almost-optimal h^(1−α−ε) rates in Lp(Ω;C([0,T];H_D^(α,q))).These rates hold for p,q∈[2,∞), α∈[−1,0], and ε>0.Source: §1 · p. 2
• The semi-discrete error also has an almost regularity-optimal h^(1−ε) rate in Lp(Ω;C([0,T];L∞(O))), the first such strong estimate for spatial semi-discretizations of stochastic Burgers.This extends prior space-time maximum-norm convergence results for semi-discrete spectral Galerkin approximations.Source: §1 · pp. 2–3
• For q≤p, weak semi-discrete error rates are (nearly) twice the corresponding strong rates; the analysis uses duality and negative-order Sobolev strong-error estimates.The authors identify these as the first weak error estimates for finite-element approximations of stochastic Burgers and in the Lq setting for spatial semi-discretizations.Source: §1 · p. 3
• The fully discrete P2 finite-element and drift-implicit Euler–Maruyama scheme has strong error estimates in the discrete space-time maximum norm when τ≤h².The estimates strengthen earlier fully discrete spectral-Galerkin results in weaker norms [38] [39].Source: §1 · p. 3
• Combining classical L2-framework techniques with discrete stochastic maximal Lp-regularity yields an Lq framework for finite-element analysis, which the authors say extends naturally to stochastic Allen–Cahn.The framework is presented as a methodological novelty for stochastic Burgers finite-element approximations.Source: §1 · pp. 3–4
• The fully discrete analysis lifts the numerical solution to continuous time and applies a Gronwall argument, using discrete stochastic maximal Lp-regularity for sharp stability estimates.The paper notes that a known approach for stochastic Navier–Stokes cannot be directly applied because the required exponential-moment bound is unavailable for stochastic Burgers.Source: §1 · p. 4

2.1 Functional analytic framework

• The functional framework uses Sobolev spaces, fractional-power domains, and real and complex interpolation spaces for the Dirichlet Laplacian.Its analytic semigroups and convolution estimates include a maximal Lp-regularity result.Source: §2.1 · pp. 5–6
• Semigroup and convolution properties provide smoothing estimates and maximal Lp-regularity used later in the analysis.The section collects these properties as preliminary lemmas.Source: §2.1 · p. 6

2.2 Stochastic framework

• The stochastic framework defines the cylindrical Wiener process, Itô integration, and γ-radonifying operators, and recalls a stochastic-integral estimate.For Hilbert-space targets, γ-radonifying operators are isometrically isomorphic to Hilbert–Schmidt operators.Source: §2.2 · p. 7

3 Regularity

• The stochastic convolution G admits a continuous modification, and the argument obtains G ∈ Lp*(Ω; C([0,T]; H_D^{θ,q})) before using the finite-measure embedding into Lp(Ω).Source: §3 · p. 8
• Stochastic maximal Lp-regularity yields H_D^{1,q} space-time regularity, with the endpoint p = 2 obtained by inclusion.The stated range includes p ≥ 2 and q ≥ 2.Source: §3 · p. 8
• For deterministic initial data in L2(O), the equation has a unique mild solution with additional regularity properties.The proposition also states an exponential moment bound.Source: §3 · p. 9
• The proof bootstraps spatial regularity using analytic-semigroup smoothing and the quadratic nonlinearity.The argument successively improves the regularity of the solution component after separating out the stochastic convolution.Source: §3 · pp. 11–12

4 Spatial semi-discretization

• Theorem 4.1 gives strong error estimates for α ∈ [−1,0] and p,q ∈ [2,∞), under its initial-data assumptions.The theorem's estimates include an arbitrarily small ε parameter; their constants are independent of h.Source: §4 · p. 14
• The P2 finite element estimates for α = 0 extend directly to the P1 method for the estimates specified in the paper.The discretization uses polynomial elements of degree at most two and a variationally defined discrete Laplacian.Source: §4 · pp. 13–14
• The additional h^−ε factor in a strong estimate arises because additive noise limits the spatial regularity of the stochastic convolution.The paper attributes the ε-loss to this intrinsic regularity gap.Source: §4 · p. 14
• The reported spatial rate h^{1−ε} mirrors the spectral Galerkin rate n^{ε−1} when n is identified with h^−1.Both parameters correspond to spatial degrees of freedom.Source: §4 · p. 15
• The weak estimate in the space-time Lp([0,T]×Ω;Lq(O)) norm has a full second-order rate, twice the corresponding strong rate.The analysis uses duality together with strong estimates in negative-order Sobolev spaces.Source: §4 · p. 15

4.1 Preliminary results

• Preliminary resolvent and generalized inverse estimates for the discrete Laplacian imply a norm equivalence.The passages refer to the resolvent estimates and resulting equivalence but do not expose their full formulas.Source: §4.1 · pp. 15–16

4.1.1 Fundamental properties of Ah and Ph

• The discrete Laplacian and projection satisfy Sobolev approximation estimates, including an extension to negative-order spaces and an L∞-error estimate.The stated bounds scale as h^(θ1−θ2); the positive-order estimate covers θ1∈[0,3] and θ2∈[0,min{1,θ1}], while the negative-order extension permits θ2∈[−3,min{1,θ1}].Source: §4.1.1 · p. 16

4.1.2 Embedding properties of the discrete spaces

• The discrete spaces have h-uniform norm equivalence with continuous Sobolev norms for θ∈[−1,1], and the Lq projection is uniformly stable for negative orders.The projection stability result covers θ∈[−1,0) and q∈(1,∞).Source: §4.1.2 · pp. 18–19
• The discrete spaces also satisfy h-uniform embeddings for p,q∈(1,∞), θ∈(0,1), and 0<α<2θ.The supplied passage states the parameter range for these embeddings but does not expose their spaces.Source: §4.1.2 · p. 19

4.1.3 Discrete analytic semigroup: smoothing, convolutions, and maximal Lp-regularity

• The discrete analytic semigroup yields smoothing estimates, and its deterministic and stochastic convolutions satisfy stability and maximal Lp-regularity estimates.The convolution results include parameter-dependent bounds; the supplied passages give conditions for some parts but omit the displayed estimates.Source: §4.1.3 · pp. 21–22

4.1.4 The spatially semi-discrete stochastic convolution Gh

• For p,q∈[2,∞) and θ∈(0,1), the semi-discrete stochastic convolution Gh has established bounds, alongside estimates comparing G and Gh for α∈[−1,0].The comparison estimates also apply for ε∈(0,1).Source: §4.1.4 · p. 23
• The convolution error G−Gh is split into a projection error and zh, which solves a deterministic evolution equation forced by AhPhG−PhAG.The resulting mild representation is used to estimate the two terms separately.Source: §4.1.4 · p. 24

Proof of

• Estimates (4.27) and (4.28) for α=0, and (4.29), also remain valid for the P1 finite element method.The paper contrasts this note with its use of the P2 method.Source: Proof of · p. 25

4.1.5 Two stability estimates

• The product-derivative estimate in Lemma 4.12 is intended to control the nonlinear convection term in subsequent error analysis.It applies for q∈(1,∞), with q′ the Hölder conjugate exponent.Source: §4.1.5 · p. 26
• Lemma 4.13 gives a stability estimate for solutions of a random integral equation, proved using an energy estimate and Gronwall’s inequality.The lemma’s assumptions include a condition involving κ0 and a bound on g; the supplied passages do not show the estimate’s full form.Source: §4.1.5 · pp. 27–29

4.2 Proof of Proposition 4.1

• The finite-dimensional stochastic differential equation has a locally Lipschitz drift with a dissipativity identity, supporting existence and uniqueness of its solution.The proof also establishes an exponential moment estimate.Source: §4.2 · p. 29
• A three-step argument establishes the stated regularity estimates by combining convolution bounds, semigroup estimates, and Sobolev embeddings.The proof concludes by applying the triangle inequality to the component estimates.Source: §4.2 · pp. 30–31

4.3 Proof of Theorem 4.1

• The proof splits the solution error into stochastic-convolution, projection, and residual components, then introduces auxiliary processes to isolate their nonlinear contributions.It defines ξ := u−G and ξ_h := u_h−G_h before constructing the auxiliary processes.Source: §4.3 · pp. 31, 33
• h^(1−α) bounds the σ_h component in the stated L^p([0,T] × Ω; H_D^(α,q)) setting for p,q ∈ [2,∞) and α ∈ [−1,0].The proof also gives h^(1−α−ε) for a related estimate and h^(1−ε) in L^p(Ω; C([0,T]; L∞(O))).Source: §4.3 · p. 33
• The proof combines the component estimates using the triangle inequality to obtain the theorem’s asserted strong error bounds.The concluding steps invoke the bounds for the auxiliary error and the σ_h component.Source: §4.3 · p. 36

4.4 Proof of Theorem 4.2

• The proof begins with boundedness estimates for the Nemytskii operator v ↦ |v|^(q−2)v, including a derivative estimate for q > 2.The derivative bound uses the chain rule and Sobolev embedding.Source: §4.4 · pp. 36–37
• For terminal L^q-norms, the proof applies the mean value theorem in a duality pairing and uses strong error estimates to obtain the weak error bound.It interpolates between u_h(T) and u(T), using their uniform regularity bounds.Source: §4.4 · pp. 37–38
• For space-time norms, the proof repeats the duality argument over [0,T] × Ω and applies the strong error estimate with α = −1.The argument uses the mean value theorem and uniform bounds on the interpolated process.Source: §4.4 · p. 38

5.2 Proof of Proposition 5.1

• For deterministic u0 ∈ W_0^{1,∞}(O), at least one fully discrete solution exists, and every solution satisfies the stated uniform bounds.The proposition also gives a bound with constants independent of h and τ.Source: §5 · p. 39

5.3 Proof of Theorem 5.1

• Under τ ≤ h^2, the scheme has strong temporal convergence rate τ^(1/2−ε) in a discrete analogue of L^p(Ω; C([0,T];L∞(O))).The estimate holds for p ∈ [2, ∞) and ε ∈ (0, 1/2), with a constant independent of h and τ.Source: §5 · p. 39
• The time-step restriction τ ≤ h^2 is described as a technical proof condition that balances temporal and spatial errors, not an essential method limitation.Under this scaling, τ^(1/2−ε) ≤ h^(1−2ε), compatible with the spatial rate O(h^(1−ε)).Source: §5 · pp. 39–40

5.1 Preliminary results

• The preliminary results establish discrete maximal-regularity estimates for implicit Euler and Euler–Maruyama, using classical semigroup estimates.The section includes deterministic and stochastic discrete estimates.Source: §5.1 · p. 40

5.3 Proof of Theorem 5.1

• The proof decomposes the error using an auxiliary continuous process, then combines its bounds with discrete maximal L^p-regularity.It defines the residual between the semidiscrete error and the auxiliary process, and bounds the auxiliary process through discrete maximal regularity.Source: §5.3 · pp. 47, 49
• Under mesh condition τ ≤ h², so τ/h ≤ τ^(1/2), (4.3) yields a bound ≲ τ^(1/2), from which the text obtains desired estimate (5.30).At t_j, e^h(t_j) = u^h(t_j) − U_j P-a.s. by (5.24).Source: §5.3 · pp. 48–49
• The mesh condition τ ≤ h^2 converts the auxiliary τ/h bound to τ^1/2, matching the τ^1/2 stochastic-integral bound.These bounds establish the auxiliary estimate (5.30).Source: §5.3 · pp. 48–49

• The spatial experiments use u0(x) = sin(πx), coefficients λn = n^-1 for 1 ≤ n ≤ 600 and zero thereafter, and 500 sample paths.For the spatial comparison, T = 0.05 and τ = 1.25×10^-5; the reference mesh has href = 2^-9.Source: §6 · pp. 49–50
• At each time step, the fully discrete implementation solves a nonlinear algebraic system using a fixed-point iteration of 40 steps.The initial iterate is the numerical solution from the preceding time step.Source: §6 · p. 50
• Strong spatial errors converge at first order, while the weak spatial error converges at a nearly quadratic rate.The errors are plotted against mesh size h on log-log scales.Source: §6 · p. 50, Figure 1 · p. 50
• The temporal strong error decays at approximately O(τ^1/2), matching the theoretical prediction when h = √τ.The reference solution uses τ* = T/48; the error is estimated from 500 independent sample paths.Source: §6 · p. 51, Figure 2 · p. 51

• The P2 spatial semi-discretization has regularity-optimal strong rates in Lp([0,T]×Ω; H_D^α,q) and an almost regularity-optimal rate in Lp(Ω; C([0,T]; L∞(O))).The paper also reports strong convergence in the space-time maximum norm.Source: §7 · p. 51
• Weak error rates for the spatial semi-discretization are nearly twice the corresponding strong rates.The paper describes these as weak error estimates in the Lq-setting.Source: §7 · p. 51
• The fully discrete P2 and drift-implicit Euler–Maruyama scheme has temporal strong rate τ^1/2−ε in the discrete space-time maximum norm when τ ≤ h^2.Source: §7 · p. 51
• The analysis combines classical L2-based techniques with discrete stochastic maximal Lp-regularity to treat the equation in general spatial Lq-spaces.The framework is expected to extend to other nonlinear stochastic partial differential equations.Source: §7 · p. 51
• Strong space-time maximum-norm and terminal-functional weak estimates for the stated stochastic Allen–Cahn problem remain future work.The paper says both problems will be addressed in forthcoming work.Source: §7 · p. 52

• When Q = 0, bound (3.8) follows trivially from (5.2); the proof then assumes Q ≠ 0.Source: A Proof of Proposition 5.1(ii) · p. 52
• The proof defines a piecewise-constant process U from the discrete solution and a continuous martingale M using its stochastic integral.The map v ↦ ⟨Q,v⟩ is bounded by the ideal property and Q ∈ γ(ℓ2,L2(O)).Source: A Proof of Proposition 5.1(ii) · p. 52
• Summing the stepwise energy estimate and dropping a nonnegative terminal term yields a bound used in the exponential-moment argument.The positive local martingale given by the Doléans-Dade exponential is a supermartingale.Source: A Proof of Proposition 5.1(ii) · pp. 52, 54
• The proof combines the intermediate estimates to obtain the desired exponential moment bound for sufficiently small τ0.The stated range is 0 < τ0 < 1/π^2.Source: A Proof of Proposition 5.1(ii) · p. 54