70,008 research outputs found
Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods
In this article, we prove convergence of the weakly penalized adaptive
discontinuous Galerkin methods. Unlike other works, we derive the contraction
property for various discontinuous Galerkin methods only assuming the
stabilizing parameters are large enough to stabilize the method. A central idea
in the analysis is to construct an auxiliary solution from the discontinuous
Galerkin solution by a simple post processing. Based on the auxiliary solution,
we define the adaptive algorithm which guides to the convergence of adaptive
discontinuous Galerkin methods
On Multiscale Methods in Petrov-Galerkin formulation
In this work we investigate the advantages of multiscale methods in
Petrov-Galerkin (PG) formulation in a general framework. The framework is based
on a localized orthogonal decomposition of a high dimensional solution space
into a low dimensional multiscale space with good approximation properties and
a high dimensional remainder space{, which only contains negligible fine scale
information}. The multiscale space can then be used to obtain accurate Galerkin
approximations. As a model problem we consider the Poisson equation. We prove
that a Petrov-Galerkin formulation does not suffer from a significant loss of
accuracy, and still preserve the convergence order of the original multiscale
method. We also prove inf-sup stability of a PG Continuous and a Discontinuous
Galerkin Finite Element multiscale method. Furthermore, we demonstrate that the
Petrov-Galerkin method can decrease the computational complexity significantly,
allowing for more efficient solution algorithms. As another application of the
framework, we show how the Petrov-Galerkin framework can be used to construct a
locally mass conservative solver for two-phase flow simulation that employs the
Buckley-Leverett equation. To achieve this, we couple a PG Discontinuous
Galerkin Finite Element method with an upwind scheme for a hyperbolic
conservation law
Wavelet Galerkin method for fractional elliptic differential equations
Under the guidance of the general theory developed for classical partial
differential equations (PDEs), we investigate the Riesz bases of wavelets in
the spaces where fractional PDEs usually work, and their applications in
numerically solving fractional elliptic differential equations (FEDEs). The
technique issues are solved and the detailed algorithm descriptions are
provided. Compared with the ordinary Galerkin methods, the wavelet Galerkin
method we propose for FEDEs has the striking benefit of efficiency, since the
condition numbers of the corresponding stiffness matrixes are small and
uniformly bounded; and the Toeplitz structure of the matrix still can be used
to reduce cost. Numerical results and comparison with the ordinary Galerkin
methods are presented to demonstrate the advantages of the wavelet Galerkin
method we provide.Comment: 20 pages, 0 figure
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