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Register a new userComputing Eigenfunctions on the Koch Snowflake: A New Grid and Symmetry
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In this paper we numerically solve the eigenvalue problem $Delta u + lambda u = 0$ on the fractal region defined by the Koch Snowflake, with zero-Dirichlet or zero-Neumann boundary conditions. The Laplacian with boundary conditions is approximated by a large symmetric matrix. The eigenvalues and eigenvectors of this matrix are computed by ARPACK. We impose the boundary conditions in a way that gives improved accuracy over the previous computations of Lapidus, Neuberger, Renka & Griffith. We extrapolate the results for grid spacing $h$ to the limit $h rightarrow 0$ in order to estimate eigenvalues of the Laplacian and compare our results to those of Lapdus et al. We analyze the symmetry of the region to explain the multiplicity-two eigenvalues, and present a canonical choice of the two eigenfunctions that span each two-dimensional eigenspace.
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We apply the Gradient-Newton-Galerkin-Algorithm (GNGA) of Neuberger & Swift to find solutions to a semilinear elliptic Dirichlet problem on the region whose boundary is the Koch snowflake. In a recent paper, we described an accurate and efficient method for generating a basis of eigenfunctions of the Laplacian on this region. In that work, we used the symmetry of the snowflake region to analyze and post-process the basis, rendering it suitable for input to the GNGA. The GNGA uses Newtons method on the eigenfunction expansion coefficients to find solutions to the semilinear problem. This article introduces the bifurcation digraph, an extension of the lattice of isotropy subgroups. For our example, the bifurcation digraph shows the 23 possible symmetry types of solutions to the PDE and the 59 generic symmetry-breaking bifurcations among these symmetry types. Our numerical code uses continuation methods, and follows branches created at symmetry-breaking bifurcations, so the human user does not need to supply initial guesses for Newtons method. Starting from the known trivial solution, the code automatically finds at least one solution with each of the symmetry types that we predict can exist. Such computationally intensive investigations necessitated the writing of automated branch following code, whereby symmetry information was used to reduce the number of computations per GNGA execution and to make intelligent branch following decisions at bifurcation points.
Let $Gamma$ be a co-compact Fuchsian group of isometries on the Poincare disk $DD$ and $Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $Delta$, equivariant by $Gamma$ with real eigenvalue $lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution $dd_{f,s}$ (the Helgason distribution) which is equivariant by $Gamma$ and supported at infinity $partialDD=SS^1$. The geodesic flow on the compact surface $DD/Gamma$ is conjugate to a suspension over a natural extension of a piecewise analytic map $T:SS^1toSS^1$, the so-called Bowen-Series transformation. Let $ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-sln |T|$. M. Pollicott showed that $dd_{f,s}$ is an eigenfunction of the dual operator $ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $psi_{f,s}$ of $ll_s$ for the eigenvalue 1, given by an integral formula [ psi_{f,s} (xi)=int frac{J(xi,eta)}{|xi-eta|^{2s}} dd_{f,s} (deta), ] oindent where $J(xi,eta)$ is a ${0,1}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface $DD/Gamma$.
We study the discretization of a Dirichlet form on the Koch snowflake domain and its boundary with the property that both the interior and the boundary can support positive energy. We compute eigenvalues and eigenfunctions, and demonstrate the localization of high energy eigenfunctions on the boundary via a modification of an argument of Filoche and Mayboroda. Holder continuity and uniform approximation of eigenfunctions are also discussed.
We give a new proof of a well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in $R^n$ with small initial data in $BMO^{-1}(R^n)$. The proof is formulated operator theoretically and does not make use of self-adjointness of the Laplacian.
We seek solutions $uinR^n$ to the semilinear elliptic partial difference equation $-Lu + f_s(u) = 0$, where $L$ is the matrix corresponding to the Laplacian operator on a graph $G$ and $f_s$ is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) {it Nonlinear Elliptic Partial Difference Equations on Graphs} (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and b) {it Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region} wherein we present some of our recent advances concerning symmetry, bifurcation, and automation fo We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newtons method, create informative graphics, and be in the underlying variational structure. We use two modified implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and Swift) to follow bifurcation branches in a robust way. By handling difficulties that arise when encountering accidental degeneracies and higher-dimension we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithms capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelis of a graph. We highlight interesting symmetry and variational phenomena.
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