2,712 research outputs found
A mixed regularization approach for sparse simultaneous approximation of parameterized PDEs
We present and analyze a novel sparse polynomial technique for the
simultaneous approximation of parameterized partial differential equations
(PDEs) with deterministic and stochastic inputs. Our approach treats the
numerical solution as a jointly sparse reconstruction problem through the
reformulation of the standard basis pursuit denoising, where the set of jointly
sparse vectors is infinite. To achieve global reconstruction of sparse
solutions to parameterized elliptic PDEs over both physical and parametric
domains, we combine the standard measurement scheme developed for compressed
sensing in the context of bounded orthonormal systems with a novel mixed-norm
based regularization method that exploits both energy and sparsity. In
addition, we are able to prove that, with minimal sample complexity, error
estimates comparable to the best -term and quasi-optimal approximations are
achievable, while requiring only a priori bounds on polynomial truncation error
with respect to the energy norm. Finally, we perform extensive numerical
experiments on several high-dimensional parameterized elliptic PDE models to
demonstrate the superior recovery properties of the proposed approach.Comment: 23 pages, 4 figure
Macro-element interpolation on tensor product meshes
A general theory for obtaining anisotropic interpolation error estimates for
macro-element interpolation is developed revealing general construction
principles. We apply this theory to interpolation operators on a macro type of
biquadratic finite elements on rectangle grids which can be viewed as a
rectangular version of the Powell-Sabin element. This theory also shows
how interpolation on the Bogner-Fox-Schmidt finite element space (or higher
order generalizations) can be analyzed in a unified framework. Moreover we
discuss a modification of Scott-Zhang type giving optimal error estimates under
the regularity required without imposing quasi uniformity on the family of
macro-element meshes used. We introduce and analyze an anisotropic
macro-element interpolation operator, which is the tensor product of
one-dimensional macro interpolation and Lagrange interpolation.
These results are used to approximate the solution of a singularly perturbed
reaction-diffusion problem on a Shishkin mesh that features highly anisotropic
elements. Hereby we obtain an approximation whose normal derivative is
continuous along certain edges of the mesh, enabling a more sophisticated
analysis of a continuous interior penalty method in another paper
Mixed finite element approximations for Darcy flow of isentropic gases
In this paper, the mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the single-phase Darcy flow of isentropic gas in a porous medium. Our numerical approach is based on the mixed finite element method (MFEM) in space, and backward-differences in time. The lowest order Raviart-Thomas elements are used. Within this frame work, we derive error estimates in suitable norms and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart- Thomas elements, are now fully exploited in the proof of convergence. Finally, we give the numerical experiments to confirm the theoretical analysis regarding convergence rates
- …