1,219 research outputs found
The DPG-star method
This article introduces the DPG-star (from now on, denoted DPG) finite
element method. It is a method that is in some sense dual to the discontinuous
Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to
solve an overdetermined discretization of a boundary value problem. In the same
vein, the DPG methodology is a means to solve an underdetermined
discretization. These two viewpoints are developed by embedding the same
operator equation into two different saddle-point problems. The analyses of the
two problems have many common elements. Comparison to other methods in the
literature round out the newly garnered perspective. Notably, DPG and DPG
methods can be seen as generalizations of and
least-squares methods, respectively. A priori error analysis and a posteriori
error control for the DPG method are considered in detail. Reports of
several numerical experiments are provided which demonstrate the essential
features of the new method. A notable difference between the results from the
DPG and DPG analyses is that the convergence rates of the former are
limited by the regularity of an extraneous Lagrange multiplier variable
Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations
The first-order system LL* (FOSLL*) approach for general second-order
elliptic partial differential equations was proposed and analyzed in [10], in
order to retain the full efficiency of the L2 norm first-order system
least-squares (FOSLS) ap- proach while exhibiting the generality of the
inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the
div-curl system with added slack vari- ables, and hence it is quite
complicated. In this paper, we apply the FOSLL* approach to the div system and
establish its well-posedness. For the corresponding finite ele- ment
approximation, we obtain a quasi-optimal a priori error bound under the same
regularity assumption as the standard Galerkin method, but without the
restriction to sufficiently small mesh size. Unlike the FOSLS approach, the
FOSLL* approach does not have a free a posteriori error estimator, we then
propose an explicit residual error estimator and establish its reliability and
efficiency bound
A first order system least squares method for the Helmholtz equation
We present a first order system least squares (FOSLS) method for the
Helmholtz equation at high wave number k, which always deduces Hermitian
positive definite algebraic system. By utilizing a non-trivial solution
decomposition to the dual FOSLS problem which is quite different from that of
standard finite element method, we give error analysis to the hp-version of the
FOSLS method where the dependence on the mesh size h, the approximation order
p, and the wave number k is given explicitly. In particular, under some
assumption of the boundary of the domain, the L2 norm error estimate of the
scalar solution from the FOSLS method is shown to be quasi optimal under the
condition that kh/p is sufficiently small and the polynomial degree p is at
least O(\log k). Numerical experiments are given to verify the theoretical
results
A mixed finite element method for nearly incompressible multiple-network poroelasticity
In this paper, we present and analyze a new mixed finite element formulation
of a general family of quasi-static multiple-network poroelasticity (MPET)
equations. The MPET equations describe flow and deformation in an elastic
porous medium that is permeated by multiple fluid networks of differing
characteristics. As such, the MPET equations represent a generalization of
Biot's equations, and numerical discretizations of the MPET equations face
similar challenges. Here, we focus on the nearly incompressible case for which
standard mixed finite element discretizations of the MPET equations perform
poorly. Instead, we propose a new mixed finite element formulation based on
introducing an additional total pressure variable. By presenting energy
estimates for the continuous solutions and a priori error estimates for a
family of compatible semi-discretizations, we show that this formulation is
robust in the limits of incompressibility, vanishing storage coefficients, and
vanishing transfer between networks. These theoretical results are corroborated
by numerical experiments. Our primary interest in the MPET equations stems from
the use of these equations in modelling interactions between biological fluids
and tissues in physiological settings. So, we additionally present
physiologically realistic numerical results for blood and tissue fluid flow
interactions in the human brain
Serendipity Face and Edge VEM Spaces
We extend the basic idea of Serendipity Virtual Elements from the previous
case (by the same authors) of nodal (-conforming) elements, to a more
general framework. Then we apply the general strategy to the case of
and conforming Virtual Element Methods, in two and three dimensions
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