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Pre-asymptotic Error Analysis of CIP-FEM and FEM for Helmholtz Equation with High Wave Number. Part II: $hp$ version

Lingxue Zhu, Haijun Wu

arXiv:1204.5061v1math.NAcs.NA

TL;DR

For Helmholtz problems at large wave numbers, pollution can dominate finite-element error bounds. This paper derives pre-asymptotic estimates for hp-FEM and CIP-FEM using modified duality, and proves unique solvability of CIP-FEM for all positive k, h, p, and penalty parameters. The estimates quantify pollution, while the penalty parameters may be tuned to reduce it.

  • Problem

    At large wave numbers, pollution can dominate the finite-element error bound, beyond the best-approximation error.

    Source: §1 · p. 2

  • Method

    The paper uses a modified duality argument to derive pre-asymptotic error estimates for hp-FEM and CIP-FEM.

    Source: §1 · pp. 2–3

  • Results

    For p=O(1), the pollution term is O(k^(2p+1)h^(2p)) under the stated solution-regularity condition, and CIP-FEM has a unique solution for any positive k, h, p, and γ.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The CIP-FEM penalty parameters may be tuned to reduce pollution relative to FEM for fixed p.

    Source: §1 · p. 2

    Extensions to PML and DtN boundary conditions are left for future work.

    Source: §1 · p. 4

Abstract

from arXiv · show

In this paper, which is part II in a series of two, the pre-asymptotic error analysis of the continuous interior penalty finite element method (CIP-FEM) and the FEM for the Helmholtz equation in two and three dimensions is continued. While part I contained results on the linear CIP-FEM and FEM, the present part deals with approximation spaces of order $p \ge 1$. By using a modified duality argument, pre-asymptotic error estimates are derived for both methods under the condition of $\frac{kh}{p}\le C_0\big(\frac{p}{k}\big)^{\frac{1}{p+1}}$, where $k$ is the wave number, $h$ is the mesh size, and $C_0$ is a constant independent of $k, h, p$, and the penalty parameters. It is shown that the pollution errors of both methods in $H^1$-norm are $O(k^{2p+1}h^{2p})$ if $p=O(1)$ and are $O\Big(\frac{k}{p^2}\big(\frac{kh}{\sigma p}\big)^{2p}\Big)$ if the exact solution $u\in H^2(\Om)$ which coincide with existent dispersion analyses for the FEM on Cartesian grids. Here $\si$ is a constant independent of $k, h, p$, and the penalty parameters. Moreover, it is proved that the CIP-FEM is stable for any $k, h, p>0$ and penalty parameters with positive imaginary parts. Besides the advantage of the absolute stability of the CIP-FEM compared to the FEM, the penalty parameters may be tuned to reduce the pollution effects.

1 Introduction

This second installment extends pre-asymptotic error analysis of CIP-FEM and FEM to hp spaces for the Helmholtz equation in two and three dimensions. It derives error estimates and establishes unconditional unique solvability for CIP-FEM, using modified duality and a separate stability argument.

1 Introduction

  • For large wave numbers, pollution can dominate the finite-element error bound; for fixed p, it can be reduced but not avoided in principle.The introduction notes that for p=1 the pollution term is O(k^3h^2), placing a tolerance-driven mesh size in the pre-asymptotic range for large k [34, 47].Source: §1 · p. 2
  • The analysis targets pre-asymptotic estimates for FEM in two and three dimensions and hp versions of both FEM and CIP-FEM, extending the linear case treated in part I [47].The paper's condition extends the asymptotic range, and the estimates apply to unstructured meshes as well as structured settings.Source: §1 · pp. 1–3
  • Both CIP-FEM and FEM admit pre-asymptotic error estimates; when p=O(1) and ||u||_{H^{p+1}(Ω)} ≲ k^p, the pollution term is O(k^{2p+1}h^{2p}), and H1 pollution bounds coincide with |k−ω|.Source: §1 · p. 3
  • CIP-FEM has a unique solution for any k, h, p, and positive γ, while its penalty parameters may be tuned to reduce pollution.The penalty parameter is chosen as a complex number in this paper and in part I [47].Source: §1 · p. 3
  • A modified duality argument uses specially designed elliptic projections to obtain L2-error estimates directly from projection errors.The analysis also relies on stability estimates for the continuous problem [39, 40].Source: §1 · p. 3
  • To prove CIP-FEM stability at higher order, the authors project a Rellich-based test function into the approximation space and control the projection difference using interface jumps.They state that the linear-method stability technique from part I [47] does not extend to higher orders because it relies on elementwise ∆uh=0.Source: §1 · p. 4
  • Extensions to PML and DtN boundary conditions are left for future work.The paper states that its theory can be extended to polyhedral domains.Source: §1 · p. 4

2 Formulation of CIP-FEM

The CIP-FEM adds jump-penalty terms to the finite element sesquilinear form, while retaining the standard FEM as the zero-penalty case. The formulation specifies the mesh, jump conventions, and penalty-parameter choice used in the analysis.

2 Formulation of CIP-FEM

  • The formulation adopts standard complex-valued L2 inner products and defines an energy space V for the sesquilinear form.The paper’s notation includes inner products on elements and boundary subsets.Source: §2 · pp. 4–5
  • The mesh uses mapped triangles or tetrahedra without hanging nodes, and h is the maximum element diameter.Element maps on shared edges or faces must induce the same parametrization.Source: §2 · p. 4
  • Interior jumps use an element-label-dependent orientation, while boundary-face normals are outward normals to the domain.The formulation also defines boundary edges or faces and uses a common penalty scale and comparable element sizes in the analysis.Source: §2 · pp. 4–5
  • The sesquilinear form is the gradient inner product plus jump-penalty terms, whose parameter is purely imaginary with positive imaginary part.The authors note that complex parameters with positive imaginary parts are also covered; real parts are set to zero for presentation and analysis convenience.Source: §2 · p. 5
  • The hp-CIP method seeks a discrete solution in Vh satisfying the variational equation, and setting all penalty parameters to zero recovers standard FEM.For an H2 solution, the penalty term vanishes, so the continuous variational equation has the stated form.Source: §2 · p. 5

3 Preliminary lemmas

The preliminary analysis combines continuous-problem stability, hp-approximation estimates, and elliptic projections. These ingredients support the later pre-asymptotic error analysis for CIP-FEM and FEM.

3 Preliminary lemmas

  • The preliminary lemmas provide continuous stability, hp-approximation bounds, and elliptic-projection error estimates for the later pre-asymptotic analysis.The section draws on continuous stability estimates and develops approximation and projection results needed in the subsequent analysis.Source: §3 · p. 6

3.1 Stability estimates of the continuous problem

  • The continuous solution decomposes into an elliptic part with an H2 bound independent of k and an oscillatory analytic part with bounds in every integer-order Hj norm.The decomposition is attributed to the stability lemma for the continuous Helmholtz problem.Source: §3.1 · p. 6

3.2 Approximation properties

  • For smooth solutions, the approximation estimates give higher-order convergence in h and p; under only H2 regularity, they give first-order convergence even for higher-order elements.The H2-based estimate uses the solution decomposition, and the authors note a slight improvement in p over earlier estimates.Source: §3.2 · pp. 6–7

3.3 Elliptic projections

  • The elliptic projections are analyzed through Galerkin orthogonality and a Nitsche duality argument, including an auxiliary solution with H1 norm bounded by the L2 error.The section combines projection approximation estimates and records error estimates between the solution and its elliptic projections.Source: §3.3 · pp. 7–8

4 Pre-asymptotic error estimates by duality argument

A modified duality argument using elliptic projections yields pre-asymptotic error estimates for both CIP-FEM and FEM. The section gives pollution bounds and identifies conditions for pollution-free behavior, with results also connected to dispersion analysis.

4 Pre-asymptotic error estimates by duality argument

  • The modified duality argument uses elliptic projections to bound the discrete solution's L2 error and derive pre-asymptotic estimates in L2 and H1 norms.The estimates apply to both CIP-FEM and FEM.Source: §4 · p. 9
  • Under condition (4.13), both CIP-FEM and FEM are pollution-free in the stated norm.The section presents (4.13) as a sufficient condition arising from the theorem's more general condition.Source: §4 · p. 10
  • The H1 pollution bounds coincide with the dispersion-analysis phase difference, while this error analysis also applies to unstructured meshes.Source: §4 · p. 12
  • For solutions in H^s(Ω), s > 2, the estimates give optimal convergence order in h and p; the H2 estimates give only first-order convergence.Source: §4 · p. 12
  • For p = O(1), if u is an oscillating solution, (4.23) shows that the pollution error in H1-norm is again bounded by O(k^(2p+1)h^(2p)).Source: §4 · p. 12

5 Stability estimates for the CIP-FEM

This section derives stability estimates for CIP-FEM and establishes that the method has a unique solution for all positive wave numbers and mesh sizes, every p ≥ 1, and γ > 0. It also explains how a projected test function addresses a difficulty in the stability analysis.

5 Stability estimates for the CIP-FEM

  • Projecting (x − xΩ) · ∇uh into the finite element space supplies a usable test function because the unprojected function is discontinuous.The projection allows the analysis to use the Rellich identity; an Oswald interpolation operator is used to analyze the projection error.Source: §5 · p. 13
  • The CIP-FEM has a unique solution for k > 0, h > 0, p ≥ 1, and γ > 0.The paper derives this well-posedness result from its stability estimates.Source: §5 · p. 18
  • For nonuniform meshes, Theorems 5.1 and 5.2 continue to hold with h replaced by the minimum element size.Source: §5 · p. 18
  • The stability result is also extended to the over-penalized CIP-FEM.Source: §5 · p. 18

6 Pre-asymptotic error estimates for the CIP-FEM by using the stability

Section 6 derives error estimates for scheme (2.4) using stability and projection estimates. It also states an extension to the over-penalized CIP-FEM.

6 Pre-asymptotic error estimates for the CIP-FEM by using the stability

  • The error-estimate derivation for scheme (2.4) combines stability estimates, projection error estimates, and linearity of the Helmholtz equation.The error is split using an elliptic projection, and the resulting discrete component is treated as a scheme (2.4) problem with source f = −k^2η.Source: §6 · p. 19
  • Theorem 6.1 assumes u ∈ H^s(Ω), s ≥ 2, and concerns the exact and discrete solutions of (1.1)–(1.2) and (2.4).The supplied theorem passages do not include its complete error bounds.Source: §6 · p. 19
  • The theorem’s stability constant is identified as C_sta, while the preceding linear-CIP-FEM result is stated for γ ≃ 1 and any k, h > 0.The theorem’s remaining estimate is not fully present in the supplied passage.Source: §6 · p. 19
  • The result is extended to the over-penalized CIP-FEM for any k, h, p > 0, with further details deferred to separate work.The supplied estimate formula is incomplete.Source: §6 · p. 19

A Approximation by hp-finite element

The appendix proves Lemmas 3.2 and 3.3.

A Approximation by hp-finite element

A.1 Proof of Lemma 3.2

The proof of Lemma 3.2 uses trace and inverse inequalities together with hp approximation properties. The stated constants depend on s but not on k, h, or p.

A.1 Proof of Lemma 3.2

  • The proof invokes trace and inverse inequalities and recalled hp approximation properties to establish Lemma 3.2.The supplied passages do not include the inequalities’ formulas.Source: §A.1 · pp. 19–20
  • The invisible constants in the two stated inequalities depend on s and are independent of k, h, and p.Source: §A.1 · p. 20
  • The remaining estimate in the proof is obtained by combining the approximation property with trace inequalities and prior estimates.The text says this establishes (3.5) and completes Lemma 3.2.Source: §A.1 · p. 20

A.2 Proof of Lemma 3.3

The proof of Lemma 3.3 builds approximants for the elliptic and analytic parts of the solution. A duality argument supports the analytic-part estimate, which improves on the cited estimate through a sharper L2 bound.

A.2 Proof of Lemma 3.3

  • Combining the H1 and L2 estimates in (A.6) gives a better upper bound than [39, Theorem 5.5], because the L2 estimate improves that proof’s bound.The remark also cites [40, Proposition 5.3] in connection with the earlier estimate.Source: §A.2 · p. 21
  • An elliptic projection of the analytic solution component is defined by requiring its sesquilinear form with every discrete test function to match that of the component.The form is a(·, ·) := (∇u, ∇v) + (u, v).Source: §A.2 · p. 22
  • A Nitsche duality argument is used to prove the analytic-part approximation estimate, with the auxiliary problem’s H2 norm bounded by the L2 approximation error.The analytic component is assumed to satisfy (3.2)–(3.3).Source: §A.2 · pp. 21–22
  • The proof combines separate approximants for the elliptic and analytic solution components, then applies the triangle inequality to obtain the combined approximation.The text says (3.7) follows by the same procedure as (3.5), omitting the details.Source: §A.2 · p. 23
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