58 research outputs found
Fast Numerical Methods for Non-local Operators
[no abstract available
Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM
Elliptic boundary value problems which are posed on a random domain can be
mapped to a fixed, nominal domain. The randomness is thus transferred to the
diffusion matrix and the loading. While this domain mapping method is quite
efficient for theory and practice, since only a single domain discretisation is
needed, it also requires the knowledge of the domain mapping.
However, in certain applications, the random domain is only described by its
random boundary, while the quantity of interest is defined on a fixed,
deterministic subdomain. In this setting, it thus becomes necessary to compute
a random domain mapping on the whole domain, such that the domain mapping is
the identity on the fixed subdomain and maps the boundary of the chosen fixed,
nominal domain on to the random boundary.
To overcome the necessity of computing such a mapping, we therefore couple
the finite element method on the fixed subdomain with the boundary element
method on the random boundary. We verify the required regularity of the
solution with respect to the random domain mapping for the use of multilevel
quadrature, derive the coupling formulation, and show by numerical results that
the approach is feasible
Multilevel preconditioning on the refined interface and optimal boundary solvers for the Laplace equation
In this paper we propose and analyze some strategies to construct asymptotically optimal algorithms for solving boundary reductions of the Laplace equation in the interior and exterior of a polygon. The interior Dirichlet or Neumann problems are, in fact equivalent to a direct treatment of the Dirichlet-Neumann mapping or its inverse i.e. the PoincarĂ©-Steklov (PS) operator. To construct a fast algorithm for the treatment of the discrete PS operator in the case of polygons composed of rectangles and regular right triangles, we apply the Bramble-Pasciak-Xu (BPX) multilevel preconditioner to the equivalent interface problem in the H1/2-setting. Furthermore, a fast matrix-vector multiplication algorithm is based on the frequency cutting techniques applied to the local Schur complements associated with the rectangular substructures specifying the nonmatching decomposition of a given polygon. The proposed compression scheme to compute the action of the discrete interior PS operator is shown to have a complexity of the order O(N logq N), q â [2,3] with memory needs of O(N log2 N) where N is the number of degrees of freedom on the polygonal boundary under consideration. In the case of exterior problems we propose a modification of the standard direct BEM whose implementation is reduced to the wavelet approximation applied to either single layer or hypersingular harmonic potentials and, in addition, to the matrix-vector multiplication for the discrete interior PS operator
Stable multilevel splittings of boundary edge element spaces
We establish the stability of nodal multilevel decompositions of lowest-order conforming boundary element subspaces of the trace space of on boundaries of triangulated Lipschitz polyhedra. The decompositions are based on nested triangular meshes created by uniform refinement and the stability bounds are uniform in the number of refinement levels. The main tool is the general theory of P.Oswald (Interface preconditioners and multilevel extension operators, in Proc. 11th Intern. Conf. on Domain Decomposition Methods, London, 1998, pp.96-103) that teaches, when stability of decompositions of boundary element spaces with respect to trace norms can be inferred from corresponding stability results for finite element spaces. -stable discrete extension operators are instrumental in this. Stable multilevel decompositions immediately spawn subspace correction preconditioners whose performance will not degrade on very fine surface meshes. Thus, the results of this article demonstrate how to construct optimal iterative solvers for the linear systems of equations arising from the Galerkin edge element discretization of boundary integral equations for eddy current problem
Compressive Wave Computation
This paper considers large-scale simulations of wave propagation phenomena.
We argue that it is possible to accurately compute a wavefield by decomposing
it onto a largely incomplete set of eigenfunctions of the Helmholtz operator,
chosen at random, and that this provides a natural way of parallelizing wave
simulations for memory-intensive applications.
This paper shows that L1-Helmholtz recovery makes sense for wave computation,
and identifies a regime in which it is provably effective: the one-dimensional
wave equation with coefficients of small bounded variation. Under suitable
assumptions we show that the number of eigenfunctions needed to evolve a sparse
wavefield defined on N points, accurately with very high probability, is
bounded by C log(N) log(log(N)), where C is related to the desired accuracy and
can be made to grow at a much slower rate than N when the solution is sparse.
The PDE estimates that underlie this result are new to the authors' knowledge
and may be of independent mathematical interest; they include an L1 estimate
for the wave equation, an estimate of extension of eigenfunctions, and a bound
for eigenvalue gaps in Sturm-Liouville problems.
Numerical examples are presented in one spatial dimension and show that as
few as 10 percents of all eigenfunctions can suffice for accurate results.
Finally, we argue that the compressive viewpoint suggests a competitive
parallel algorithm for an adjoint-state inversion method in reflection
seismology.Comment: 45 pages, 4 figure
Simulating gravitational waves passing through the spacetime of a black hole
We investigate how GWs pass through the spacetime of a Schwarzschild black
hole using time-domain numerical simulations. Our work is based on the
perturbed 3+1 Einstein's equations up to the linear order. We show explicitly
that our perturbation equations are covariant under infinitesimal coordinate
transformations. Then we solve a symmetric second-order hyperbolic wave
equation with a spatially varying wave speed. As the wave speed in our wave
equation vanishes at the horizon, our formalism can naturally avoid boundary
conditions at the horizon. Our formalism also does not contain coordinate
singularities and, therefore, does not need regularity conditions. Then, based
on our code, we simulate both finite and continuous initially plane-fronted
wave trains passing through the Schwarzschild black hole. We find that for the
finite wave train, the wave zone of GWs is wildly twisted by the black hole.
While for the continuous wave train, unlike geometric optics, GWs can not be
sheltered by the back hole. A strong beam and an interference pattern appear
behind the black hole along the optical axis. Moreover, we find that the
back-scattering due to the interaction between GWs and the background curvature
is strongly dependent on the direction of the propagation of the trailing
wavefront relative to the black hole.Comment: 24 pages, 9 figure
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
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