Computing eigenfunctions on the Koch Snowflake: A new grid and symmetry

Neuberger, John M.; Sieben, Nándor; Swift, James W.

doi:10.1016/j.cam.2005.03.075

Cited by 23 publications

(34 citation statements)

References 7 publications

(20 reference statements)

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“…Consider the PDE (1) on the cube, with f s odd. For positive integers p, q, and r, not all 1, the set (11) A p,q,r = span ψ i,j,k | i p , j q , k r ∈ Z is an AIS. The function space A 1,1,1 is all of V , so it is not an AIS.…”

Section: Symmetry Of Functionsmentioning

confidence: 99%

Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

Neuberger¹,

Sieben²,

Swift³

2013

SIAM J. Appl. Dyn. Syst.

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Abstract. We seek discrete approximations to solutions u : Ω → R of semilinear elliptic partial differential equations of the form ∆u + fs(u) = 0, where fs is a one-parameter family of nonlinear functions and Ω is a domain in R d . The main achievement of this paper is the approximation of solutions to the PDE on the cube Ω = (0, π) 3 ⊆ R 3 . There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace.This article extends the authors' work in Automated Bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs (Int. J. of Bifurcation and Chaos, 2009), wherein they combined symmetry analysis with modified implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and Swift) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our paper An MPI Implementation of a Self-Submitting Parallel Job Queue (Int. J. of Parallel Prog., 2012).

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Section: Symmetry Of Functionsmentioning

confidence: 99%

Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

Neuberger¹,

Sieben²,

Swift³

2013

SIAM J. Appl. Dyn. Syst.

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“…The matrix that approximates the Laplacian operator with boundary conditions on the region is used in the finite difference method. The graph Laplacian matrix gives the approximation for the Laplacian operator with zero Neumann boundary conditions [34], so the synchrony subspaces of exo-balanced partitions are invariant. For other boundary conditions, the subspaces that are invariant under the matrix approximating the Laplacian include the synchrony subspaces of balanced partitions.…”

Section: 7mentioning

confidence: 99%

Invariant Synchrony Subspaces of Sets of Matrices

Neuberger¹,

Sieben²,

Swift³

2020

SIAM J. Appl. Dyn. Syst.

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A synchrony subspace of R n is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices form a lattice. Applications of these invariant synchrony subspaces include equitable and almost equitable partitions of the vertices of a graph used in many areas of graph theory, balanced and exo-balanced partitions of coupled cell networks, and coset partitions of Cayley graphs. We study the basic properties of invariant synchrony subspaces and provide many examples of the applications. We also present what we call the split and cir algorithm for finding the lattice of invariant synchrony subspaces. Our theory and algorithm is further generalized for non-square matrices. This leads to the notion of tactical decompositions studied for its application in design theory.2000 Mathematics Subject Classification. 15A72, 34C14, 34C15, 37C80, 06B23, 90C35, 1 0 0 5 6 0 1 0 7 8 0 0 1 7 8 0 0 0 0 0 of [P (A) | Q] has leading columns in the first three columns. So Col(Q) ⊆ Col(P (A)) = sys(A). Proposition 6.4. Let M = {M 1 , . . . , M m } ⊆ R m×n , A ∈ Π(m), and B ∈ Π(n). Then M l sys(B) ⊆ sys(A) for all l if and only if A ≤ ψ( M 1 P (B) · · · M m P (B) ). Proof. First assume that M l sys(B) ⊆ sys(A) for all l. Then Col(M l P (B)) = M l Col(P (B)) ⊆ Col(P (A))for all l. Proposition 6.1 (3) and (1) implyNow assume that A ≤ ψ( M 1 P (B) · · · M m P (B) ). Then A ≤ ψ(M l P (B)) by Proposition 6.1(2). Hence Col(M l P (B)) ⊆ sys(A) by Lemma 6.2.The following result is the special case of Proposition 6.4 when m = n and B = A. Coarsest invariant refinementIn this section we develop an algorithm that finds the coarsest invariant partition that is not larger than a given partition.Proposition 7.1. If M ⊆ R n×n and A ∈ Π(n), then (Π M (n)∩ ↓ A) is in Π M (n).Proof. Note that the down-set ↓ A is taken in Π(n). Proposition 3.5 implies thatThe following is a generalization of [49, Theorem 5]. It is a recursive process for finding cir M (A), so we refer to it as the cir algorithm. Proposition 7.3. Let M = {M 1 , . . . , M m } ⊆ R n×n . Given a partition A 0 := A ∈ Π(n), recursively define A k+1 := ψ( P (A k ) Q k ), where Q k := M 1 P (A k ) · · · M m P (A k ) .There is a k 0 such that A k = cir M (A) for all k ≥ k 0 .

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“…The exposition of the symmetry properties of the region was crucial in choosing an appropriate initial guess, since the bifurcation branches of crossing and avoided crossing are by their very nature close together. The article [11] and the forthcoming article [12] use more advanced techniques similar to ours to study the linear and the nonlinear problem on the Koch Snowflake region, a region with a fractal boundary.…”

Section: Discussionmentioning

confidence: 99%

“…The sorting of eigenfunctions and eigenvalues relies on projections; the deeper underlying structure involves representation theory (see [11]). …”

Section: Symmetry Of Eigenfunctionsmentioning

confidence: 99%

GNGA for general regions: Semilinear elliptic PDE and crossing eigenvalues

Hineman

Neuberger

2007

Communications in Nonlinear Science and Numerical Simulation

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Abstract. We consider the semilinear elliptic PDE ∆u + f (λ, u) = 0 with the zero-Dirichlet boundary condition on a family of regions, namely stadions. Linear problems on such regions have been widely studied in the past. We seek to observe the corresponding phenomena in our nonlinear setting. Using the Gradient Newton Galerkin Algorithm (GNGA) of Neuberger and Swift, we document bifurcation, nodal structure, and symmetry of solutions. This paper provides the first published instance where the GNGA is applied to general regions. Our investigation involves both the dimension of the stadions and the value λ as parameters. We find that the so-called crossings and avoided crossings of eigenvalues as the dimension of the stadions vary influences the symmetry and variational structure of nonlinear solutions in a natural way.

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Computing eigenfunctions on the Koch Snowflake: A new grid and symmetry

Cited by 23 publications

References 7 publications

Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

Invariant Synchrony Subspaces of Sets of Matrices

GNGA for general regions: Semilinear elliptic PDE and crossing eigenvalues

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