50 research outputs found
Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric Finite Element Method
Let \Omega \subset \RR^d, , be a bounded domain with
piecewise smooth boundary and let be an open subset of a
Banach space . Motivated by questions in "Uncertainty Quantification," we
consider a parametric family of uniformly strongly
elliptic, second order partial differential operators on . We
allow jump discontinuities in the coefficients. We establish a regularity
result for the solution u: \Omega \times U \to \RR of the parametric,
elliptic boundary value/transmission problem , , with
mixed Dirichlet-Neumann boundary conditions in the case when the boundary and
the interface are smooth and in the general case for . Our regularity and
well-posedness results are formulated in a scale of broken weighted Sobolev
spaces \hat\maK^{m+1}_{a+1}(\Omega) of Babu\v{s}ka-Kondrat'ev type in
, possibly augmented by some locally constant functions. This implies
that the parametric, elliptic PDEs admit a shift theorem that
is uniform in the parameter . In turn, this then leads to
-quasi-optimal rates of convergence (i.e. algebraic orders of convergence)
for the Galerkin approximations of the solution , where the approximation
spaces are defined using the "polynomial chaos expansion" of with respect
to a suitable family of tensorized Lagrange polynomials, following the method
developed by Cohen, Devore, and Schwab (2010)
Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D
Let be a potential on \RR^3 that is smooth everywhere except at a
discrete set \maS of points, where it has singularities of the form
, with for close to and continuous on
\RR^3 with for p \in \maS. Also assume that and
are smooth outside \maS and is smooth in polar coordinates around each
singular point. We either assume that is periodic or that the set \maS is
finite and extends to a smooth function on the radial compactification of
\RR^3 that is bounded outside a compact set containing \maS. In the
periodic case, we let be the periodicity lattice and define \TT :=
\RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the
eigenfunctions of the Schr\"odinger-type operator acting on
L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by
restricting the action of to Bloch waves. Under some additional
assumptions, we extend these regularity and solvability results to the
non-periodic case. We sketch some applications to approximation of
eigenfunctions and eigenvalues that will be studied in more detail in a second
paper.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55
(103), no. 2/201
Multigrid methods for saddle point problems: Oseen system
We develop and analyze multigrid methods for the Oseen system in fluid flow. We show that the W-cycle algorithm is a uniform contraction if the number of smoothing steps is sufficiently large. Numerical results that illustrate the performance of the methods are also presented
Analysis of Schrodinger operators with inverse square potentials II: FEM and approximation of eigenfunctions in the periodic case
In this article, we consider the problem of optimal approximation of eigenfunctions of Schrödinger operators
with isolated inverse square potentials and of solutions to equations involving such operators. It is known in
this situation that the finite element method performs poorly with standard meshes. We construct an alter-
native class of graded meshes, and prove and numerically test optimal approximation results for the finite
element method using these meshes. Our numerical tests are in good agreement with our theoretical results
Analysis of Schrodinger operators with inverse square potentials I: regularity results in 3D
Let V be a potential on R3 that is smooth everywhere except at a discrete set
S of points, where it has singularities of the form Z/ 2, with (x) = |x − p| for x close to p
and Z continuous on R3 with Z(p) > −1/4 for p 2 S. Also assume that and Z are smooth
outside S and Z is smooth in polar coordinates around each singular point. We either assume
that V is periodic or that the set S is finite and V extends to a smooth function on the radial
compactification of R3 that is bounded outside a compact set containing S. In the periodic
case, we let be the periodicity lattice and define T := R3/ . We obtain regularity results in
weighted Sobolev space for the eigenfunctions of the Schr¨odinger-type operator H = − + V
acting on L2(T), as well as for the induced k–Hamiltonians Hk obtained by restricting the
action of H to Bloch waves. Under some additional assumptions, we extend these regularity
and solvability results to the non-periodic case. We sketch some applications to approximation
of eigenfunctions and eigenvalues that will be studied in more detail in a second paper