The Cauchy problem of f(R) gravity

The initial value problem of metric and Palatini f(R)gravity is studied by using the dynamical equivalence between these theories and Brans-Dicke gravity. The Cauchy problem is well-formulated for metric f(R)gravity in the presence of matter and well-posed in vacuo. For Palatini f(R)gravity, instead, the Cauchy problem is not well-formulated.Comment: 16 latex pages, to appear in Class. Quantum Grav; typographical errors corrected, new references adde

Discrete maximum principle for higher-order finite elements in 1D.

We formulate a sufficient condition on the mesh under which we prove the discrete maximum principle (DMP) for the one-dimensional Poisson equation with Dirichlet boundary conditions discretized by the hp-FEM. The DMP holds if a relative length of every element K in the mesh is bounded by a value H* (p) € [0.9, 1], where p ≥ 1 is the polynomial degree of the element K. The values H* (p) are calculated for 1 ≤ p ≤ 100. © 2007 American Mathematical Society

Discrete conservation of nonnegativity for elliptic problems solved by the hp-FEM.

Most results related to discrete nonnegativity conservation principles (DNCP) for elliptic problems are limited to finite differences and lowest-order finite element methods (FEM). In this paper we show that a straightforward extension to higher-order finite element methods (hp-FEM) in the classical sense is not possible. We formulate a weaker DNCP for the Poisson equation in one spatial dimension and prove it using an interval computing technique. Numerical experiments related to the extension of this result to 2D are presented. © 2007 IMACS

Modular hp-FEM system HERMES and its application to Maxwell's equations.

In this paper, we introduce a multi-physics modular hp-FEM system HERMES. The code is based on a novel approach where the finite element technology (mesh processing and adaptation, numerical quadrature, assembling and solution of the discrete problems, a-posteriori error estimation, etc.) is fully separated from the physics of the solved problems. The physics is represented via simple modules containing PDE-dependent parameters as well as hierarchic higher-order finite elements satisfying the conformity requirements imposed by the PDE. After describing briefly the modular structure of HERMES and some of its functionality, we focus on its application to the time-harmonic Maxwell's equations. We present numerical results which illustrate the capability of the hp-FEM to reduce both the number of degrees of freedom and the CPU time dramatically compared to standard lowest-order FEM. © 2007 IMACS

Imposing orthogonality to hierarchic higher-order finite elements.

We propose a new class of hierarchic higher-order finite elements suitable for the hp-FEM discretization of symmetric linear elliptic problems. These elements use shape functions which are partially orthonormal on the reference domain under the energetic inner product induced by the elliptic problem. We present numerical experiments showing excellent conditioning properties of the new partially orthogonal shape functions compared to other popular sets of hierarchic shape functions. © 2007 IMAC