136 research outputs found
Finite Element Approximations for Stokes-Darcy Flow with Beavers-Joseph Interface Conditions
Numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits. Such situations arise, for example, for groundwater flows in karst aquifers. the coupled flow is modeled by the Darcy equation in a porous medium and the Stokes equations in the conduit domain. on the interface between the matrix and conduit, Beavers-Joseph interface conditions, instead of the simplified Beavers-Joseph-Saffman conditions, are imposed. Convergence and error estimates for finite element approximations are obtained. Numerical experiments illustrate the validity of the theoretical results. © 2010 Society for Industrial and Applied Mathematics
A Virtual Element Method for a Nonlocal FitzHugh-Nagumo Model of Cardiac Electrophysiology
We present a Virtual Element Method (VEM) for a nonlocal reaction-diffusion
system of the cardiac electric field. To this system, we analyze an
-conforming discretization by means of VEM which can make use of
general polygonal meshes. Under standard assumptions on the computational
domain, we establish the convergence of the discrete solution by considering a
series of a priori estimates and by using a general compactness
criterion. Moreover, we obtain optimal order space-time error estimates in the
norm. Finally, we report some numerical tests supporting the theoretical
results
Space-time Methods for Time-dependent Partial Differential Equations
Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space.
Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations
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
Longer time accuracy for incompressible Navier-Stokes simulations with the EMAC formulation
In this paper, we consider the recently introduced EMAC formulation for the
incompressible Navier-Stokes (NS) equations, which is the only known NS
formulation that conserves energy, momentum and angular momentum when the
divergence constraint is only weakly enforced. Since its introduction, the EMAC
formulation has been successfully used for a wide variety of fluid dynamics
problems. We prove that discretizations using the EMAC formulation are
potentially better than those built on the commonly used skew-symmetric
formulation, by deriving a better longer time error estimate for EMAC: while
the classical results for schemes using the skew-symmetric formulation have
Gronwall constants dependent on with the Reynolds
number, it turns out that the EMAC error estimate is free from this explicit
exponential dependence on the Reynolds number. Additionally, it is demonstrated
how EMAC admits smaller lower bounds on its velocity error, since {incorrect
treatment of linear momentum, angular momentum and energy induces} lower bounds
for velocity error, and EMAC treats these quantities more accurately.
Results of numerical tests for channel flow past a cylinder and 2D
Kelvin-Helmholtz instability are also given, both of which show that the
advantages of EMAC over the skew-symmetric formulation increase as the Reynolds
number gets larger and for longer simulation times.Comment: 21 pages, 5 figure
Finite element approximation for a delayed generalized Burgers-Huxley equation with weakly singular kernels: Part I Well-posedness, Regularity and Conforming approximation
The analysis of a delayed generalized Burgers-Huxley equation (a non-linear
advection-diffusion-reaction problem) with weakly singular kernels is carried
out in this work. Moreover, numerical approximations are performed using the
conforming finite element method (CFEM). The existence, uniqueness and
regularity results for the continuous problem have been discussed in detail
using the Faedo-Galerkin approximation technique. For the numerical studies, we
first propose a semi-discrete conforming finite element scheme for space
discretization and discuss its error estimates under minimal regularity
assumptions. We then employ a backward Euler discretization in time and CFEM in
space to obtain a fully-discrete approximation. Additionally, we derive a prior
error estimates for the fully-discrete approximated solution. Finally, we
present computational results that support the derived theoretical results
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