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Stability analysis of boundary element methods for the wave equation: The weakly singular operator in $1d$
Daniel Hoonhout, Richard Löscher, Carolina Urzúa-Torres
TL;DR
Discrete stability for wave-equation boundary element methods depends on trial and test spaces whose inf-sup condition remains uniform under mesh refinement. The paper derives optimal pairings for the one-dimensional weakly singular operator, characterizes the test space under a mesh condition, and connects stability to how well propagated waves are resolved.
Problem
Discrete boundary element stability requires trial and test spaces with an inf-sup condition independent of mesh refinement.
Source: §1 · p. 1Method
The paper applies an abstract optimal-test-space framework to the one-dimensional weakly singular wave operator and characterizes its optimal test space.
Source: §1 · p. 1, §3.2 · p. 13Results
The optimal test space is explicitly characterized under a mesh condition, whose restriction reflects the need to resolve waves traveling from the left boundary to the right.
Source: §3.2 · p. 13Takeaways & Limitations
If the mesh does not resolve the propagated function well enough, discrete stability gets worse.
Source: §5.2 · p. 28The explicit optimal-test-space characterization relies on a mesh restriction; without it, stability is expected when arriving waves are approximated sufficiently well.
Source: §3.2 · p. 13
Abstract
from arXiv · showhide
We discuss the construction of stable discrete pairs of trial and test spaces for the wave equation. Building on a general abstract operator-theoretic framework, we show how optimal pairings can be derived for a one-dimensional boundary element formulation of the wave equation with prescribed Dirichlet data, based on an indirect single layer potential ansatz. In this setting, the optimal test space can be computed analytically and has a physical interpretation, reflecting the non-local nature of the underlying operator.
1 Introduction
Discrete stability for wave-equation boundary element methods requires trial and test spaces whose inf-sup condition remains uniform under mesh refinement. This paper applies optimal-test-space analysis to the one-dimensional weakly singular operator and examines classical low-order Lagrangian boundary elements.
1 Introduction
- Discrete stability requires trial and test spaces whose inf-sup condition holds independently of mesh refinement.The continuous variational formulations are continuous and inf-sup stable, but only occasionally elliptic,.Source: §1 · p. 1
- The paper uses the optimal test-space framework to analyze stable and unstable pairings for the one-dimensional weakly singular wave operator.The analysis focuses on classical low-order Lagrangian boundary elements; related operators, including the double layer operator K, are also noted [9].Source: §1 · p. 1
- The analysis characterizes optimal test spaces, presents mesh-constrained stability results, and uses numerical examples to examine their sharpness.The paper then extends the stability analysis to a setting without the mesh constraint.Source: §1 · p. 1
2 Abstract Setting
The abstract framework links trial and test spaces through the Riesz operator to construct stable discrete formulations. It establishes solvability and quasi-optimality results, explains practical test-space choices, and relates a discrete residual to error estimation.
2 Abstract Setting
- Using the Riesz operator, the transformed test space R_Y^-1 B(X) yields an elliptic formulation that guides stable discrete-space design.The abstract setting also establishes unique solvability for the variational formulation.Source: §2 · p. 3
2.1 The optimal and the practical test space
- The optimal test space inherits the continuous inf-sup constant, but may be difficult to construct; practical test spaces allow flexibility in their dimension relative to the trial space.In particular, equal trial and test dimensions are not required for unique solvability with the practical test space.Source: §2.1 · pp. 4–5
2.2 Error estimates
- Under the discrete inf-sup condition, the practical formulation has a quasi-optimal error estimate, differing from the optimal-space estimate only in its constant.The theorem establishes unique solvability of the practical formulation under that condition.Source: §2.2 · p. 7
2.3 Error estimator
- The discrete residual p_h can serve as an error estimator when the discrete inf-sup condition holds.The estimator result assumes that condition for the trial and test spaces.Source: §2.3 · p. 8
3 Application to the Weakly Singular Operator for the Wave Equation in One Dimension
This section applies the abstract stability framework to the one-dimensional weakly singular wave-equation operator, analyzing stable and unstable discrete pairings and characterizing optimal test spaces. It establishes uniform stability for a refined test-space choice with piecewise-constant trials and identifies an optimal integrated test space under the mesh assumption.
3.1 Properties of the Weakly Singular Operator
- The operator splits into diagonal and cross-boundary terms; the latter couple time slices, while the diagonal action is slice-independent.The cross-boundary terms vanish before time L, and the stated shifting relation equates the diagonal and off-diagonal contributions on adjacent slices when T = nL.Source: §3.1 · p. 10
3.2 Characterization of the Optimal Test Space
- Under Assumption 3.1, the optimal test space is explicitly characterized, and the discrete formulation has a unique solution with a stability estimate.The mesh alignment ensures waves arriving at the opposite boundary after the first time slice are resolved accurately; stability may still hold without this constraint if arriving waves are approximated sufficiently well.Source: §3.2 · pp. 12–13
3.3.1 Why you shouldn’t date yourself: The unstable pair S0(Σh) −S0(Σh)
- The equal piecewise-constant trial and test pair S0(Σh) is unstable, with its discrete inf-sup stability failing for any mesh-independent constant.The paper proves instability previously reported in the literature [7, Chap. 2.1], using highly oscillatory functions to expose the failure.Source: §3.3.1 · pp. 13, 15
3.3.2 Why someone more refined could be the answer: The stable pair S0(Σh)− S0(Σh/m), m ∈N≥2
- With Assumption 3.1, using S0(Σh/m) as the test space for S0(Σh) yields stability for every integer refinement factor m ≥ 2.The result improves the previously stated sufficient condition m ≥ 3 and provides a sharp discrete inf-sup constant.Source: §3.3.2 · p. 18
- For m≥2, the refined pair has c(m)^2 = 1 − 1/m^2 and satisfies the discrete inf-sup condition uniformly.The spaces are Yh = S0(Σh/m) and Xh = S0(Σh).Source: §3.3.2 · p. 21
- Because the refined trial and test spaces have different dimensions, the saddle-point formulation adds degrees of freedom but provides ph as an error estimator.Source: §3.3.2 · p. 21
3.3.3 Choose an integrated version of yourself: The stable pair S0(Σh)−S1
- Under mesh Assumption 3.1, Xh = S0(Σh) has optimal test space Yhopt = S1, forming another discrete inf-sup stable pair.Source: §3.3.3 · p. 21
4 Numerical experiments
The numerical experiments examine the sharpness of two stable trial-test pairs, with results on mesh refinement and final-time dependence.
4.1 The stable pair S0(Σh) −S1
- At fixed T, the discrete inf-sup constant for S0(Σh) −S1 does not asymptotically depend on h, confirming discrete stability.Figure 4a plots the constant against mesh-element count.Source: §4.1 · p. 22, Figure 4 · p. 23
- As T increases, the S0(Σh) −S1 inf-sup constant decreases but remains strictly positive; the uniform case matches αV is(n), while the non-uniform case closely resembles it.Figure 4b reports the final-time comparison.Source: §4.1 · p. 22, Figure 4 · p. 23
4.2 The stable pair S0(Σh) −S0(Σh/2)
- For S0(Σh) −S0(Σh/2), the non-uniform-mesh inf-sup constant decays when the mesh fails Assumption 3.1; comparisons across ending times use uniform refinements.Fixed-ending-time results also include an adaptive mesh; stability is shown only for meshes satisfying Assumption 3.1.Source: §4.2 · p. 23, Figure 5 · p. 24
5 Stability With Weaker Mesh Constraints
The section seeks discrete stability without the original mesh constraint, first using an ellipticity-based estimate and then refining the analysis through projection errors and approximate mesh alignment. The refined result identifies a sufficient alignment condition for n^-1 behavior, while experiments show that condition is not necessary.
5 Stability With Weaker Mesh Constraints
- The section motivates relaxing mesh restrictions by noting that composing the weakly singular operator with a suitable operator can yield ellipticity and mesh-independent stability.It then considers the double layer potential as a route toward stability without the mesh assumption.Source: §5 · p. 24
5.1 A dull stability estimate without mesh constraints
- The arbitrary-mesh stability estimate is not sharp: experiments indicate n^-1 behavior, while the estimate's constant has qualitatively different behavior.The section introduces a projection-based refinement to improve the estimate.Source: §5.1 · p. 25
5.2 Choose someone who lives up to their projection: stability under a different mesh condition
- If the mesh resolves the propagated information sufficiently well, the inf-sup constant behaves like n^-1; if it does not resolve the propagated function well enough, discrete stability gets worse.Source: §5.2 · pp. 28–29
5.2.1 Approximated compatibility
- Condition (5.8) requires shifted nodes to lie within O(n^-1) of opposite-boundary mesh nodes relative to local mesh size; exact alignment is sufficient but not required.The paper presents this relative alignment as a sufficient condition for the desired asymptotic scaling.Source: §5.2.1 · pp. 30–31
- Numerical experiments show that meshes violating condition (5.8) can still yield an inf-sup constant behaving like n^-1, so the condition is sufficient but not necessary.Figure 6 presents results for (4.1) on meshes violating (5.8).Source: §5.2.1 · p. 31, Figure 6 · p. 32
A Auxiliary results for non-matching meshes
The appendix develops interpolation estimates for non-matching partitions. Its auxiliary result bounds interpolation error for continuous piecewise-linear functions under stated partition assumptions.
A Auxiliary results for non-matching meshes
- The estimates provide technical building blocks needed for Theorem 5.6.The appendix identifies these interpolation estimates as prerequisites for that theorem.Source: A Auxiliary results for non-matching… · p. 33
- The appendix establishes interpolation identities for piecewise-linear functions, including exact interpolation outside the element containing a shifted node.The argument treats the case where the node lies inside an element and derives the local interpolation expression.Source: A Auxiliary results for non-matching… · p. 35
- Lemma A.3 gives an interpolation-error bound for continuous piecewise-linear functions on non-matching partitions satisfying Assumption (A.4).The partition assumptions bound the number of interior nodes of one partition within each element of the other and constrain shifted-node distance relative to local mesh size.Source: A Auxiliary results for non-matching… · pp. 35–36