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Polytopic symmetric interior-penalty approximation of elliptic SPDEs driven by spatial white noise

Tristan Pryer

arXiv:2609.06082v1math.NAcs.NA

TL;DR

Low spatial regularity prevents the usual SIP consistency argument for elliptic SPDEs driven by white noise. The paper instead restricts the same continuum forcing to the discontinuous space, proves sharp strong-convergence results including for reentrant corners, and establishes optimality with a finite-dimensional lower bound.

  • Problem

    The white-noise-driven solution has low spatial regularity, so the usual SIP consistency argument based on elementwise gradients and normal traces need not be available.

    Source: §1 · pp. 1–2

  • Method

    The analysis canonically restricts the same continuum white-noise functional to the discontinuous polynomial space and splits the error into forcing-restriction and deterministic SIP components.

    Source: §1 · pp. 1–2

  • Results

    A matching finite-dimensional lower bound establishes optimality of the sharp strong-convergence rates for fixed-degree approximation spaces of comparable dimension, while the reentrant-corner case retains a sharp first-order stochastic rate.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The stochastic approximation retains its sharp first-order rate on two-dimensional polytopic domains with a single reentrant corner, despite the deterministic loss of elliptic regularity.

    Source: §1 · p. 2

    The stated problem setting is limited to bounded polytopic Lipschitz domains in dimensions two and three.

    Source: §2 · p. 2

Abstract

from arXiv · show

We analyse symmetric interior-penalty discontinuous Galerkin approximations of elliptic stochastic partial differential equations driven by spatial Gaussian white noise on quasi-uniform polytopic meshes. The low regularity of the exact random field precludes the usual SIP consistency argument. We instead restrict the same continuum white-noise functional canonically to the discontinuous polynomial space and separate the error into stochastic forcing-restriction and deterministic discretisation components. For every fixed polynomial degree $k\geq1$, we prove the sharp two-sided estimate \[ \|Y-Y_h\|_{L^2(\Xi;L^2(\Omega))} \simeq h^{2-d/2}, \] giving rates $h$ in two dimensions and $h^{1/2}$ in three dimensions. A finite-dimensional lower bound establishes optimality. The analysis allows polytopic elements with arbitrarily many and arbitrarily small faces, including admissible non-convex elements. In two dimensions we also treat polygons with a single reentrant corner. Although the deterministic approximation exhibits an angle-dependent loss of regularity, the singular resolvent contribution is finite rank, and the sharp first-order stochastic rate is retained. The discontinuous structure yields an exact element-local sampler for the restricted white noise. Numerical experiments on irregular polygonal, non-convex polytopic and reentrant domains support the predicted convergence behaviour.

1. Introduction

The paper develops a sharp strong-convergence analysis for SIP approximations driven by the same continuum white noise, despite the solution’s low regularity. It also treats reentrant corners and broad polytopic mesh geometries.

1. Introduction

  • Because the exact field is too irregular for the usual SIP consistency argument, the analysis restricts the same white-noise functional to the discontinuous polynomial space and separates forcing-restriction from deterministic SIP error.The deterministic term is an ordinary SIP error with an L2(Ω) right-hand side.Source: §1 · p. 2
  • A matching finite-dimensional lower bound establishes optimality of the sharp strong convergence rates for approximation spaces of comparable dimension.The first main result concerns fixed polynomial degree on quasi-uniform polytopic meshes.Source: §1 · p. 2
  • For domains with a single reentrant corner, the stochastic approximation retains the sharp first-order rate despite the deterministic loss of elliptic regularity.The proof uses the finite-dimensional structure of the corner singularity, which does not alter the leading stochastic approximation error.Source: §1 · p. 2
  • The analysis covers polytopic meshes whose elements may have arbitrarily many or small faces and may be non-convex, under a uniform star-shapedness condition.The estimates are uniform with respect to the number and relative sizes of element faces.Source: §1 · p. 2
  • The discontinuous formulation permits exact sampling of the restricted white-noise load using independent element-local Gaussian vectors.The construction also gives exact couplings across agglomerated mesh hierarchies.Source: §1 · p. 2
  • Experiments test the predicted rates on regular two- and three-dimensional domains and assess first-order stochastic convergence across reentrant corner angles.The study also examines irregular polygonal and polytopic mesh families.Source: §1 · p. 2

2. Problem setup

The problem setup defines a Dirichlet elliptic operator on a bounded polytopic domain and formulates its white-noise-driven solution through the resolvent. The white-noise solution is L2-valued in the stated dimensions, with Sobolev regularity depending on dimension.

2. Problem setup

  • The setting is a bounded polytopic Lipschitz domain in dimension two or three, with a complete probability space for the random fields.Source: §2 · p. 2
  • The positive self-adjoint operator L is the Dirichlet realization of −∆+κ^2, and the problem is defined by LY=W.Source: §2 · p. 3
  • The principal analysis assumes an elliptic H2 regularity bound; on reentrant polygons, the paper replaces this assumption with an explicit corner-singularity decomposition.Source: §2 · p. 3

2.2. Spatial white noise and the elliptic solution.

  • Compactness gives an L2-orthonormal eigensystem for L, whose eigenvalues satisfy the stated Weyl-law growth bounds.Source: §2 · p. 3
  • The resolvent defines a unique L2(Ω)-valued white-noise solution in dimensions two and three, with available Sobolev regularity H1−ε in two dimensions and H1/2−ε in three.The spectral convergence criterion is finite for s<2−d/2; at s=2−d/2 it has critical logarithmic divergence.Source: §2.2 · pp. 3–4

3. Polytopic symmetric interior-penalty discretisation

The section constructs a SIP discretisation on polytopic meshes, defines canonical white-noise restriction, and separates stochastic forcing restriction from deterministic resolvent approximation. It establishes a sharp two-sided strong L2 error rate for the resulting approximation.

3. Polytopic symmetric interior-penalty discretisation

  • The analysis splits the error into white-noise restriction and deterministic SIP approximation components.This lets each part be treated with tools appropriate to stochastic forcing restriction or an L2 right-hand side.Source: §3 · p. 4

3.1. Polytopic meshes and discrete spaces.

  • The mesh condition permits arbitrarily small faces and arbitrarily many faces per element, while projection estimates have constants independent of face count and relative sizes.The elementwise L2 projection error is bounded by h^2 times the H2 norm.Source: §3.1 · p. 5

3.4. SIP form and discrete operator. ¶

  • The SIP form is stable and its continuity constants are independent of h and of the number and relative sizes of element faces.The discrete operator defined by the form is symmetric positive definite.Source: §3.4 · p. 6

3.6. Deterministic resolvent approximation.

  • The deterministic resolvent approximation has L2 error bounded by h^2 times the L2 norm of the forcing.The bound is for the SIP solution with an L2 right-hand side under the stated regularity and stability assumptions.Source: §3.6 · p. 7

3.8. Canonical discrete white noise.

  • The canonical restricted load has an exact element-local sampler because element load vectors are mutually independent Gaussian vectors.The coefficient vector on each element is obtained by solving the local mass-matrix system.Source: §3.8 · p. 8

3.9. Restricted noise and strong convergence.

  • The strong error obeys ch^(2−d/2) ≤ ||Y−Yh||L2(Ξ;L2(Ω)) ≤ Ch^(2−d/2), yielding the sharp strong L2 rate h in two dimensions and h^(1/2) in three dimensions.Source: §3.9 · p. 10

4. Extension to a reentrant polygon

For a two-dimensional polygon with a single reentrant corner, the deterministic corner loss persists, but the sharp first-order stochastic rate remains. The proof separates the regular resolvent error from finite-rank corner contributions.

4. Extension to a reentrant polygon

  • The extension assumes one reentrant corner with angle ω⋆ ∈ (π, 2π), convex remaining vertices, and boundary-conforming meshes with x⋆ as a mesh vertex.The penalty parameter is chosen sufficiently large for Lemma 3.5 to hold.Source: §4 · p. 10

4.1. Corner structure and deterministic SIP estimates.

  • The corner decomposition writes Tf as an H2-regular part plus a rank-one singular term, and the regular resolvent has singular values σj(Treg) ≲ j^-1.The rank-one perturbation leaves the stated singular-value bound for Treg applicable to the full resolvent.Source: §4.1 · p. 11
  • The uniform whole-boundary trace estimate for ∇s⋆ requires no lower bound on any individual face size.The argument uses π/ω⋆ > 1/2 to choose an admissible exponent.Source: §4.1 · p. 12

4.3. Hilbert–Schmidt resolvent error.

  • ch ≤ ∥Y − Yh∥L2(Ξ;L2(Ω)) ≤ Ch on the stated reentrant polygons, retaining the sharp first-order strong rate.The lower bound follows because Yh takes values in Vh and Nh ≃ h^-2.Source: §4.3 · p. 15
  • The infinite-dimensional regular part has Hilbert–Schmidt error O(h), while the two rank-one corner corrections are o(h) because ω⋆ < 2π.This decomposition explains why the deterministic regularity loss does not change the stochastic rate.Source: §4.3 · pp. 13, 15

5. Numerical experiments

The experiments test the predicted dimension-dependent continuum error on regular polytopic meshes and the first-order stochastic rate across reentrant corner angles. They report both convergence results alongside the estimators and mesh families used to evaluate them.

5.2. Experiment 1: regular polytopic meshes.

  • On the unit square and cube, an explicit Dirichlet eigensystem estimates continuum strong error; the experiments compare it with the predicted dimension-dependent rate.Figure 3 covers four two-dimensional mesh families against O(h); Figure 5 covers regular and jittered hexahedral and non-convex polyhedral families against O(h1/2).Source: §5 · p. 15, Two dimensions · p. 16, Figure 3 · p. 17

5.3. Experiment 2: stochastic reentrant problem.

  • For reentrant domains, a family of corner angles is tested using a coupled fine-level reference to assess whether the first-order stochastic rate persists.The stochastic hierarchy preserves the reentrant vertex at each refinement level and includes many-sided and non-convex elements.Source: §5 · p. 15, §5.3 · pp. 18–19

5.1. Continuum strong-error estimator on (0, 1)d.

  • The continuum strong-error estimator uses the explicit Dirichlet eigensystem on the unit square and cube, while reentrant-domain convergence uses a coupled fine-level reference.These provide the two numerical comparisons: dimension-dependent continuum error and stochastic convergence across corner angles.Source: §5 · p. 15

5.3. Experiment 2: stochastic reentrant problem.

  • The experiments recover the h^(2-d/2) rate on regular polygonal and polyhedral domains and the first-order stochastic rate across the reentrant angle sweep.The reentrant-domain comparison uses a coupled fine-level reference.Source: §5 · p. 15, §5.3 · p. 19

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