Quadrature by fundamental solutions: kernel-independent layer potential evaluation for large collections of simple objects

Abstract

Well-conditioned boundary integral methods for the solution of elliptic boundary value problems (BVPs) are powerful tools for static and dynamic physical simulations. When there are many close-to-touching boundaries (e.g., in complex fluids) or when the solution is needed in the bulk, nearly singular integrals must be evaluated at many targets. We show that precomputing a linear map from surface density to an effective source representation renders this task highly efficient, in the common case where each object is “simple”, i.e., its smooth boundary needs only moderately many nodes. We present a kernel-independent method needing only an upsampled smooth surface quadrature, and one dense factorization, for each distinct shape. No (near-)singular quadrature rules are needed. The resulting effective sources are drop-in compatible with fast algorithms, with no local corrections nor bookkeeping. Our extensive numerical tests include 2D FMM-based Helmholtz and Stokes BVPs with up to 1000 objects (281000 unknowns), and a 3D Laplace BVP with 10 ellipsoids separated by 1/30 of a diameter. We include a rigorous analysis for analytic data in 2D and 3D.

Appendices

Appendix: 1. Proofs of main theorems on correctness of QFS for analytic data

Here we prove Theorems 2, 4 and 5. We use ideas from Doicu–Eremin–Wriedt [25, Ch. IV, Thms. 2.1-2], who considered only pure SLP for Helmholtz for d = 3. We combine the cases of d = 2 (curves) and d = 3 (surfaces), so for simplicity use the latter term for both. All three theorems involve a collocation (check) surface γ_c whose interior Ω_c contains Ω, the interior of the user’s layer potential surface ∂Ω. This handles both QFS-B and QFS-D cases. In addition there is a QFS source surface γ ⊂Ω whose interior we denote by Ω₋. All three surfaces γ_c, ∂Ω, and γ are smooth and non-self-intersecting. We abbreviate u_n := ∂u/∂n.

Proof of Theorem 2 (Laplace)

Using u to also denote the continuation of the solution, one may read off its data on γ, and the exterior Green’s representation formula [46, (1.4.5)] holds,

$$u({\mathbf{x}}) = {\int}_{\gamma} \biggl[ -G({\mathbf{x}},{\mathbf{y}}) u_{\mathbf{n}}({\mathbf{y}}) + \frac{\partial G(\mathbf{x},\mathbf{y})}{\partial \mathbf{n}_{\mathbf{y}}} u(\mathbf{y}) \biggr] ds_{\mathbf{y}}~, \qquad \mathbf{x} \in \mathbb{R}^d\backslash\overline{{\Omega}_-}~.$$

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Note that, by the decay condition, no constant term is needed. Let v be the unique solution to the Laplace BVP interior to γ with Dirichlet data v = u on γ, then let $v_{{\mathbf {n}}}^{-}$ be its normal derivative, then the exterior extinction GRF holds

$$0 = {\int}_{\gamma} \biggl[ G(\mathbf{x},\mathbf{y}) v_{\mathbf{n}}^-(\mathbf{y}) - \frac{\partial G({\mathbf{x}},{\mathbf{y}})}{\partial {\mathbf{n}}_{\mathbf{y}}} v({\mathbf{y}}) \biggr] ds_{\mathbf{y}}~, \qquad {\mathbf{x}} \in \mathbb{R}^d\backslash\overline{{\Omega}_-}~.$$

(71)

Adding the last two equations cancels the DLP terms, leaving (17) with

$$\sigma({\mathbf{y}}) := v_{{\mathbf{n}}}^{-}({\mathbf{y}}) - u_{{\mathbf{n}}}({\mathbf{y}})~,\qquad {\mathbf{y}}\in\gamma~,$$

a density solving (16). Since all data on γ was analytic, σ is certainly smooth.

For uniqueness, instead let σ solve (16) with zero RHS. Then let u be given by (17) with this σ; by construction u vanishes on γ_c. By uniqueness of the exterior Dirichlet BVP for d = 3, or by Lemma 3 for d = 2, then u ≡ 0 throughout $\mathbb {R}^d\backslash {\Omega }$, so that u_n ≡ 0 on γ_c. Then by unique continuation from the Cauchy data u ≡ u_n ≡ 0 on γ_c, u vanishes throughout $\mathbb {R}^{d}\backslash \overline {{\Omega }_{-}}$. Since the potential is continuous across a single-layer [59, Thm. 6.14], and u is harmonic in Ω₋, then u solves the interior Dirichlet BVP in Ω₋ with vanishing data. By uniqueness of this BVP, u vanishes in Ω₋, thus both limits of u_n on either side of γ vanish, so by the jump relation [59, Thm. 6.18], σ ≡ 0.

Finally, the Laplace solution (17) reproduces u in the statement of the theorem, since it matches in value on γ_c, and again one uses uniqueness of the exterior Dirichlet BVP for d = 3, or Lemma 3 for d = 2.

Proof of Theorem 4 (Helmholtz)

We use u to denote the continuation of u as a Helmholtz solution onto γ. Let v solve the homogeneous Helmholtz impedance BVP in the interior of γ, with boundary data

$$v_{\mathbf{n}}-i\eta v = u_{\mathbf{n}}-i\eta u \qquad \text{ on } {\gamma}.$$

(72)

It is standard that this BVP has a unique solution for any real k [86, Sec. 8.8] (or [30, Prop 2.1]). Adding the Helmholtz versions of the GRFs (68) (which applies since u is radiative [19, Sec. 2.2]) and (69), and adding and subtracting iη(v − u), we get

$$u({\mathbf{x}}) = {\int}_{\gamma} \left[ G({\mathbf{x}},{\mathbf{y}})[v_{{\mathbf{n}}}-i \eta v - (u_{{\mathbf{n}}}-i \eta u) + i\eta (v-u)] - \frac{\partial G(\mathbf{x},\mathbf{y})}{\partial \mathbf{n}_{\mathbf{y}}} (v-u) \right] ds_{\mathbf{y}}$$

which, by (70) simplifies to give (24) with σ := (u − v)|_γ. Choosing x ∈ γ_c shows that σ solves (23).

Uniqueness follows by similar arguments to the Laplace case: instead let σ solve (23) with zero RHS, then let u be given by (24). Then u vanishes on γ_c, so by the uniqueness of the exterior Dirichlet BVP (19)–(21), u also vanishes throughout $\mathbb {R}^{d}\backslash {\Omega }_{c}$. Since u is analytic [19, Thm. 2.2], by unique continuation u also vanishes in $\mathbb {R}^{d}\backslash \overline {{\Omega }_{-}}$. By the jump relations its interior limits of u on γ are u⁻ = −σ and $u^{-}_{{\mathbf {n}}} = -i\eta \sigma$. Thus $u_{{\mathbf {n}}}^{-}-i\eta u^{-} = 0$ on γ, and by construction u is also a Helmholtz solution in Ω₋. By the uniqueness of the impedance BVP in Ω₋, then u ≡ 0 in Ω₋, so, again by either jump relation, σ ≡ 0.

Finally, (24) reproduces u in the statement of the theorem since both are radiative Helmholtz solutions matching in value on γ_c, and one invokes uniqueness for the radiative exterior Dirichlet BVP.

Proof of Theorem 5 (Stokes)

The proof is as for Helmholtz but with −iη replaced by 1. The Green’s representation formulae (68)–(69) apply for the Stokes velocity field, with traction data T(u,p) (defined, e.g., in [46, Sec. 2.3.1]) in place of normal derivative data, and D(x,y) from (30) in place of the scalar kernel ∂G(x,y)/∂n_y. Then let (v,q) solve the homogeneous Stokes BVP interior to γ, with Robin (“impedance”) data

$$\mathbf{T}(\mathbf{v},q) + \mathbf{v} = {\mathbf{T}}(\mathbf{u},p) + \mathbf{u} \qquad \text{ on }\gamma~.$$

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A solution exists by Lemma 25 below. Adding (68) (which applies since u has a zero constant term), and (69), and adding and subtracting v −u, we get

$$\mathbf{u}({\mathbf{x}}) = {\int}_{\gamma} \left[ G({\mathbf{x}},{\mathbf{y}})[{\mathbf{T}}(\mathbf{v},q)+\mathbf{v} - ({\mathbf{T}}(\mathbf{u},p)+\mathbf{u}) - (\mathbf{v}-\mathbf{u})] - D(\mathbf{x},\mathbf{y}) (\mathbf{v}-\mathbf{u}) \right] ds_{\mathbf{y}}$$

which, by (71) simplifies to give (32) with σ := (u −v)|_γ. Choosing x ∈ γ_c shows that σ solves (31). This completes existence. The remainder is similar to Laplace, apart from the following. One needs uniqueness for the exterior Stokes Dirichlet BVP with zero constant term: in d = 2 Lemma 3 (logarithmic capacity condition) is replaced by the nonsingularity hypothesis for γ_c in the theorem statement. The unique continuation argument relies on each component of u being analytic [62, p. 60]. The rest of the proof is as for Helmholtz, replacing −iη by 1, with the uniqueness of the interior Robin BVP assured by Lemma 25 below.

Lemma 25 (Existence and uniqueness for Stokes interior Robin BVP)

Let ${\Omega }\subset \mathbb {R}^{d}$, be bounded with smooth boundary ∂Ω. Let $\mathbf {f}: \partial {\Omega }\to \mathbb {R}^{d}$ be given smooth data. Then the problem to find a vector field v and scalar function q solving the Stokes equations − μΔv + ∇q = 0 and ∇⋅v = 0 in Ω, with Robin data T(v,q) + v = f on ∂Ω, has at most one solution. In addition, if d = 3, or d = 2 and the 2 × 2 matrix mapping Σ to ω in (28) is nonsingular, then it has exactly one solution.

Proof

Uniqueness follows easily as in [47, p. 83] or [79, p.51]. One uses (v,q) for both the solution pairs in Green’s 1st identity [62, p. 53] to get

$$\frac{\mu}{2}{\int}_{\Omega} \| \nabla \mathbf{v} + \nabla \mathbf{v}^{T} \|_{F}^{2} = {\int}_{{\partial{\Omega}}} {\mathbf{T}}(\mathbf{v},q)\cdot \mathbf{v} ds~,$$

where F indicates the Frobenius norm of the d × d tensor. Applying the Robin condition with f ≡0 shows that the right-hand side is non-positive, so that both vanish, so that v ≡0. For existence, suppose that ϕ ∈ C(∂Ω)^d solves the BIE

$$(S + D^T + 1/2)\boldsymbol{\phi} = \mathbf{f}$$

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where D^T is the adjoint double-layer operator. Then $\mathbf {v} = \mathcal { S}\boldsymbol {\phi }$ solves the Stokes equations in Ω with the correct Robin data following from the jump relations, thus is a solution. By the Fredholm alternative, to prove existence for (72), one may prove uniqueness for the adjoint BIE (S + D + 1/2)ψ = 0. This is already known in d = 3 [40, Thm 2.1]. In d = 2 the constant term again rears its ugly head [47], but given the hypothesis we can prove uniqueness as follows. Let ψ solve the homogeneous adjoint BIE, then construct ${\mathbf {w}} = (\mathcal {S} + \mathcal {D}){\boldsymbol {\psi }}$ and r the corresponding pressure representation, which solve the modified exterior BVP (25)–(28) with given ω = 0, but Σ arbitrary. By the 2 × 2 matrix nonsingularity hypothesis the exterior solution is unique, hence trivial. By the jump relations on ∂Ω, the interior limits are w⁻ = ψ and T(w,r)⁻ = −ψ, so that (w,r) solves the interior Robin BVP with zero data T(w,r)⁻ + w⁻ = 0. By uniqueness proved above, the solution is identically zero, so again by the jump relations, ψ ≡0.

We suspect that there is a way to remove the above Stokes domain nonsingularity condition in d = 2, perhaps following [47].

Appendix: 2. Geometry generation for large-scale 2D examples

Here we present an algorithm to generate K simple polar-Fourier shapes ∂Ω_i, $i=1,\dots ,K$, located at randomly generated centers, that obey the distance and variation criteria of Section 4.1. Its inputs are ${d_{\min \limits }}$, a body radius scale r₀, and a routine randomcenter that returns fresh centers c.

First we make a list of centers $\{\mathbf {c}_{1},\dots ,\mathbf {c}_{K}\}$ that are far enough apart. Starting with the empty list,

• 1. Generate a new candidate center c via randomcenter,

• 2. Append c to the list if c has distance at least 2r₀ from all c_i in the list,

• 3. Repeat 1-2 until the list has K centers.

The body ∂Ω_i is now chosen from the star-shaped family defined about the center c_i by the polar parameterization $r(t)=r_{0}(1+a\cos \limits (ft + \phi ))$, where a is the “wobble” amplitude, f its frequency, and ϕ its rotation. For example Fig. 1a shows c = 0, r₀ = 1, a = 0.3, f = 5, ϕ = 0.2. f is drawn randomly from $\{3,4,\dots ,7\}$ with a distribution function {16/31, 8/31, 4/31, 2/31, 1/31}, to include higher frequencies less often. The amplitude a is uniform random in [0, 0.3(3/f)^3/2]. Thus higher frequencies will tend to have smaller amplitudes, in order to prevent any single boundary from dominating the resolution requirements. ϕ is uniform random in [0, 2π). Since the maximum radius of a body is currently (1 + a)r₀, intersections are possible, and there may not be any bodies that are $\approx {d_{\min \limits }}$ apart. Thus we use the following to adjust all body radii:

• 1. All geometries are rescaled so that the farthest distance from center to boundary is $r_{0}+{d_{\min \limits }}/2$. At this point, no boundaries can intersect and all boundaries must be separated by at least ${d_{\min \limits }}$.

• 2. 10% of the geometries are chosen at random, and for each chosen geometry:

  • (a) Denote the current maximum radius of the geometry by R.

  • (b) (expansion) The geometry is rescaled to increase its radius by ${d_{\min \limits }}$, and minimal separation distances between the geometry and all its nearest neighbors are computed.

  • (c) Step (b) is repeated until either the geometries current radius is > 1.5R, or the geometry is separated from a nearest neighbor by $<{d_{\min \limits }}$.

  • (d) If the prior step is terminated because the geometry is $<{d_{\min \limits }}$ from a nearest neighbor, proceed to the next step; otherwise handling for this geometry is finished.

  • (e) (rescue) The geometry is rescaled to decrease its radius by ${d_{\min \limits }}/10$, and separation distances between the geometry and all its nearest neighbors are computed.

  • (f) Step (e) is repeated until the radius of the geometry is between $({d_{\min \limits }}, 1.1 {d_{\min \limits }}]$.

• 3. Step 2 is repeated three times.

• 4. For speed, the prior items are computed using approximate methods (distances are computed pointwise over barely resolved boundaries), and in rare instances boundaries may be closer together than ${d_{\min \limits }}$. A final rescue step is performed for every boundary with upsampled boundaries and using full Newton iterations to compute the minimal distances.

Although elaborate, this process allows us to efficiently place K polydisperse boundaries in a specified manner throughout a domain, with a separation no less than ${d_{\min \limits }}$ between the individual boundaries. When the initial set of boundaries are packed sufficiently tightly, the expansion steps always produce at least some boundaries whose expansion is terminated because they are too close to others; thus, due to how the rescue stage is implemented, there will always be some close pairs of boundaries separated by between ${d_{\min \limits }}$ and $1.1 {d_{\min \limits }}$.