Structure preserving approximation of dissipative evolution problems

Abstract

We present a general abstract framework for the systematic numerical approximation of dissipative evolution problems. The approach is based on rewriting the evolution problem in a particular form that complies with an underlying energy or entropy structure. Based on the variational characterization of smooth solutions, we are then able to show that the approximation by Galerkin methods in space and discontinuous Galerkin methods in time automatically leads to numerical schemes that inherit the dissipative behavior of the evolution problem. The proposed framework is rather general and can be applied to a wide range of applications. This is demonstrated by a detailed discussion of a variety examples ranging from diffusive partial differential equations to Hamiltonian and gradient systems.

Comparison of a finite-element and finite-volume scheme for a degenerate cross-diffusion system for ion transport

“…Next, let φ ∈ (H 2 (Ω) ∩ H 1 D (Ω)) n . As we cannot use φ i directly as a test function in (11), we take I h φ ∈ S D (T h ) n , where I h is the interpolation operator, see (10). In order to pass to the limit in (11), we rewrite the integral involving the diffusion matrix:…”

“…In [23], an implicit Euler Galerkin approximation in entropy variables for a Poisson-Maxwell-Stefan system was shown to converge. Recently, an abstract framework for the numerical approximation of evolution problems with entropy structure was presented in [10]. The discretization is based on a discontinuous Galerkin method in time and a Galerkin approximation in space.…”

“…. The H 2 regularity of the initial and boundary data ensures that the standard interpolation converges to the given data, see (10) below.…”

A Partitioned Finite Element Method for the Structure-Preserving Discretization of Damped Infinite-Dimensional Port-Hamiltonian Systems with Boundary Control

“…Our main concern is to provide a numerical method that preserves, at the discrete level, the geometrical structure of the original controlled PDE; for short, we look for a structure-preserving method which automatically transforms the Stokes-Dirac structure into a finite-dimensional Dirac structure: in the last decade, quite a number of ways have been explored, see e.g. [20,26,13,10,19]. Recently in [4], a method based on the weak formulation of the Partial Differential Equation and the use of the celebrated Finite Element Method has emerged.…”

A two-level iterative scheme for general sparse linear systems based on approximate skew-symmetrizers

“…Our main motivation to study identity-plus-skew-symmetric preconditioners are linear systems arising in the time discretization of dissipative Hamiltonian differential equations of the form Eż = (J − R) z + f (t), z(t 0 ) = z 0 , whereż denotes the derivative with respect to time, J is a skew-symmetric matrix, R is symmetric positive semidefinite, and E is the symmetric positive semidefinite Hessian of a quadratic energy functional (Hamiltonian) H(z) = 1 2 z T Ez; see, e.g., [2,10,14,20,31,32] for such systems in different physical domains and applications. If one discretizes such ETNA Kent State University and Johann Radon Institute (RICAM) SPARSE SHIFTED SKEW-SYMMETRIZERS AND PRECONDITIONING 371 systems in time, e.g., with the implicit Euler method, and sets z k = z(t k ), then in each time step t k one has to solve a linear system of the form (1.1) (E − h(J − R))z k+1 = Ez k + hf (t k ).…”