9 research outputs found
On dual Schur domain decomposition method for linear first-order transient problems
This paper addresses some numerical and theoretical aspects of dual Schur
domain decomposition methods for linear first-order transient partial
differential equations. In this work, we consider the trapezoidal family of
schemes for integrating the ordinary differential equations (ODEs) for each
subdomain and present four different coupling methods, corresponding to
different algebraic constraints, for enforcing kinematic continuity on the
interface between the subdomains.
Method 1 (d-continuity) is based on the conventional approach using
continuity of the primary variable and we show that this method is unstable for
a lot of commonly used time integrators including the mid-point rule. To
alleviate this difficulty, we propose a new Method 2 (Modified d-continuity)
and prove its stability for coupling all time integrators in the trapezoidal
family (except the forward Euler). Method 3 (v-continuity) is based on
enforcing the continuity of the time derivative of the primary variable.
However, this constraint introduces a drift in the primary variable on the
interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte
stabilization to limit this drift and we derive bounds for the stabilization
parameter to ensure stability.
Our stability analysis is based on the ``energy'' method, and one of the main
contributions of this paper is the extension of the energy method (which was
previously introduced in the context of numerical methods for ODEs) to assess
the stability of numerical formulations for index-2 differential-algebraic
equations (DAEs).Comment: 22 Figures, 49 pages (double spacing using amsart
Non-negative mixed finite element formulations for a tensorial diffusion equation
We consider the tensorial diffusion equation, and address the discrete
maximum-minimum principle of mixed finite element formulations. In particular,
we address non-negative solutions (which is a special case of the
maximum-minimum principle) of mixed finite element formulations. The discrete
maximum-minimum principle is the discrete version of the maximum-minimum
principle.
In this paper we present two non-negative mixed finite element formulations
for tensorial diffusion equations based on constrained optimization techniques
(in particular, quadratic programming). These proposed mixed formulations
produce non-negative numerical solutions on arbitrary meshes for low-order
(i.e., linear, bilinear and trilinear) finite elements. The first formulation
is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative
constraint to the variational statement of the Raviart-Thomas formulation. The
second non-negative formulation based on the variational multiscale
formulation.
For the former formulation we comment on the affect of adding the
non-negative constraint on the local mass balance property of the
Raviart-Thomas formulation. We also study the performance of the active set
strategy for solving the resulting constrained optimization problems. The
overall performance of the proposed formulation is illustrated on three
canonical test problems.Comment: 40 pages using amsart style file, and 15 figure
Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus
We derive a numerical method for Darcy flow, hence also for Poisson's equation in first order form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is its discretization on simplicial complexes such as triangle and tetrahedral meshes. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. Our method requires the use of meshes in which each simplex contains its circumcenter. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solution in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this paper. We also include a discussion of the boundary condition in terms of exterior calculus