Fully discrete stochastic maximal regularity and $H^\infty$-calculus for second-order elliptic operators

1. Introduction

• Earlier work established stochastic maximal regularity for spatial or temporal discretizations, while the cited fully discrete Laplacian result used implicit Euler and had a suboptimal maximal estimate.Stochastic maximal regularity is used in numerical analysis for stability estimates and convergence rates for SPDEs.Source: §1 · p. 2

1.1. Main results.

• The discrete spatial operator A_h has a bounded H∞-calculus of angle zero with a constant independent of h.This result provides the foundation for the paper’s fully discrete stochastic maximal-regularity results.Source: §1.1 · p. 4
• The paper characterizes discrete fractional spaces through real interpolation with equivalent norms uniformly in h.The characterization identifies spaces appearing in the estimates with standard Sobolev and Besov spaces.Source: §1.1 · p. 4
• The fully discrete estimates establish stochastic maximal regularity and a sharp maximal estimate, with constants independent of h and τ, for the stated range of p and q.The paper treats second-order elliptic operators and a broad class of temporal schemes; the stated main results focus on operators without lower-order terms.Source: §1 · p. 2, §1.1 · pp. 3, 5
• The spatial semi-discretization also satisfies stochastic maximal regularity, with constants independent of h.The paper presents this as a byproduct of the fully discrete analysis.Source: §1.1 · p. 5

Overview.

• The H∞-calculus proof treats the principal part first, then handles the full operator by a perturbation argument; the paper also presents mathematical background and notation.The principal-part analysis is developed before the perturbation argument for the full operator.Source: Overview · p. 5, Notation · p. 6

2.1. Sectorial operators, functional calculus, and interpolation.

• Sectoriality is defined by spectral containment and a resolvent bound, and sectorial operators of angle below π/2 generate bounded analytic semigroups.The preliminaries also rely on standard real and complex interpolation results.Source: §2.1 · p. 6

2.1.1. Functional calculus.

• The H∞-calculus extends the sectorial H1-calculus to bounded analytic functions, while bounded H∞-calculus implies bounded imaginary powers.On Hilbert spaces, generators of contraction semigroups have bounded H∞-calculus; on Lq, positive contraction semigroups provide another sufficient class.Source: §2.1.1 · pp. 6–7, §2.1.2 · p. 7

2.1.2. Fractional powers.

• For sectorial operators with bounded imaginary powers, fractional-power domains are identified with complex interpolation spaces.The text states this identification as Lemma 2.1 for interpolation parameters θ in (0, 1).Source: §2.1.2 · p. 7

2.2. Properties of the operator A.

• The elliptic operator A is defined from a form on a bounded, convex C2 domain in R3, with bounded measurable coefficients and a symmetric elliptic principal part.The preliminaries develop sectoriality, invertibility, and fractional-space properties for its Lq realizations.Source: §2.2 · p. 7

2.3. Finite elements.

• The finite-element space uses continuous piecewise-linear functions on a conforming quasi-uniform tetrahedral mesh, with boundary vertices on the domain boundary.The preliminaries also give a global inverse estimate and state Lq stability and convergence properties for the orthogonal projection.Source: §2.3 · pp. 8–9

2.4. Stochastic integration.

• Stochastic integration is formulated using γ-radonifying operators, which reduce to Hilbert–Schmidt operators for Hilbert spaces and identify with Lq(O; H) when X=Lq(O).A cylindrical Brownian motion is represented through independent scalar Brownian motions along an orthonormal basis, rather than as an H-valued random variable.Source: §2.4 · p. 9

2.5. (Discrete) stochastic maximal regularity.

• Stochastic maximal Lp-regularity controls the solution of the stochastic evolution equation, and a bounded H∞-calculus of angle below π/2 is sufficient for this regularity.The result applies for p>2, with p=2 allowed when q=2, and includes a maximal estimate for the mild solution.Source: §2.5 · p. 10
• The discrete-time framework considers exponential and A-stable consistent rational approximations, including implicit Euler and sub-diagonal Padé functions, on admissible uniform time grids.The text notes prior equivalence between discrete-in-time and continuous-in-time stochastic maximal regularity [14].Source: §2.5 · p. 11
• Discrete stochastic maximal regularity requires an admissible constant uniform in the time step τ, although it may depend on T.The definition uses spatial regularity shifted by 1/2 because the discrete-to-continuous norm equivalence for A_h holds only through α = 1/2.Source: §2.5 · p. 11
• Continuous stochastic Lp-maximal regularity and discrete stochastic ℓp-maximal regularity are equivalent under Theorem 2.9.The theorem also relates their regularity constants.Source: §2.5 · p. 11
• A bounded H∞-calculus for A with angle below π/2 suffices for R to have discrete stochastic ℓp-maximal regularity.The result applies for p > 2, and also for p = 2 when q = 2.Source: §2.5 · pp. 11–12
• The equivalence and discrete maximal estimate also hold with appropriate time weights, and schemes using A(θ)-stable rational functions are allowed when θ exceeds A’s sectoriality angle.The permitted stability angle satisfies θ ∈ (ω(A), π/2].Source: §2.5 · p. 12

3. H∞-calculus for the discrete operator Ah

• For q ∈ (1, ∞) and ν ∈ (0, π/2), λ0 + Ah has a bounded H∞(Σν)-calculus on Xq_0,h and is invertible, with λ0 and the calculus bound independent of h.The shift may depend on q, the coefficients, and ν; when b = c = 0, λ0 can be zero.Source: §3 · p. 12
• A perturbation argument extends the principal-part calculus to Ah, with the shift λ0 absorbing the lower-order terms.The operator is decomposed as Ah = A#_h + Bh, where Bh discretizes the lower-order terms.Source: §3 · p. 12

3.1. H∞-calculus for the principal part A#

• For q ∈ (1, ∞), the principal-part operator A#_h admits a bounded H∞-calculus of angle zero, uniformly in h.The result assumes Assumption 1.1 for A#; the stated theorem applies for every ν ∈ (0, π/2).Source: §3.1 · pp. 13, 15
• The principal-part calculus yields uniform-in-h characterizations of discrete fractional spaces and certain real interpolation scales used in fully discrete stochastic maximal regularity estimates.The passages state fractional-space results for α ranges including [0, 1/2] and [0, 1], and a real interpolation characterization.Source: §3.1 · p. 13

3.1.2. Proof of Theorem 3.3 and its corollaries.

• Jh has a bounded H∞-calculus of angle zero whose constant equals that of A#q and is independent of h.The proof also establishes resolvent estimates uniform in h for every ν ∈ (0, π/2).Source: §3.1.2 · p. 17
• The fractional-space norm characterization in Corollary 3.4 requires Ritz-projection stability, although its necessity for Theorem 3.3 is unknown.The proof obtains the characterization through stability and interpolation arguments.Source: §3.1.2 · pp. 19–20

3.2. H∞-calculus for the full operator Ah.

• The full discrete operator and its shifted version admit bounded H∞(Σν)-calculi with constants independent of h.The proof extends the principal-part results to lower-order terms by perturbation.Source: §3.2 · p. 21
• The full-operator discrete fractional spaces are determined by the principal part, with uniform norm equivalences for α ∈ [0, 1].The equivalence compares the inhomogeneous norms of the principal and full operators.Source: §3.2 · p. 21
• The principal-part comparison yields equivalent real interpolation norms between the continuous and discrete spaces, uniformly in h.Corollary 3.16 gives this comparison for θ ∈ (0, 1) and p ∈ [1, ∞].Source: §3.2 · pp. 21–22

• For the full discretization on a bounded convex domain in R3, r is either e−z or an A-stable rational function consistent of order ℓ≥1 with r(∞)=0.The principal part A# = −∇·a∇ satisfies Assumption 1.1, and λ0≥0 is large enough that λ0 + Ah admits a bounded H∞ calculus uniformly in the mesh size h. If the lower-order coefficients bi = c = 0, λ0 may be 0.Source: §4 · p. 22
• For q∈[2,∞) and p∈(2,∞), allowing p=2 when q=2, the fully discrete estimates hold with constants independent of h and τ.The estimates also permit their right-hand sides to be bounded using the norm of the forcing term g.Source: §4 · p. 22
• The spatial semi-discretization also satisfies stochastic maximal regularity with a constant independent of h.Its estimate likewise permits a right-hand side bounded by the norm of g.Source: §4 · pp. 22–23
• The section also records deterministic discrete-in-space maximal regularity, with the solution norms bounded by the forcing norm uniformly in h.The cited discussion notes this estimate is established in the literature and identifies prior proofs with differing coefficient and boundary-condition settings.Source: §4 · p. 23