Discontinuous hp-Finite Element Methods for Advection-Diffusion Problems

We consider the hp-version of the discontinuous Galerkin finite element method for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second--order elliptic and parabolic equations, first-order hyperbolic equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hp-optimal error bound is derived. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by half a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments

Experimental and numerical investigation of Helmholtz resonators and perforated liners as attenuation devices in industrial gas turbine combustors

This paper reports upon developments in the simulation of the passive control of combustion dynamics in industrial gas turbines using acoustic attenuation devices such as Helmholtz resonators and perforated liners. Combustion instability in gas turbine combustors may, if uncontrolled, lead to large-amplitude pressure fluctuations, with consequent serious mechanical problems in the gas turbine combustor system. Perforated combustor walls and Helmholtz resonators are two commonly used passive instability control devices. However, experimental design of the noise attenuation device is time-consuming and calls for expensive trial and error practice. Despite significant advances over recent decades, the ability of Computational Fluid Dynamics to predict the attenuation of pressure fluctuations by these instability control devices is still not well validated. In this paper, the attenuation of pressure fluctuations by a group of multi-perforated panel absorbers and Helmholtz resonators are investigated both by experiment and computational simulation. It is demonstrated that CFD can predict the noise attenuation from Helmholtz resonators with good accuracy. A porous material model is modified to represent a multi-perforated panel and this perforated wall representation approach is demonstrated to be able to accurately predict the pressure fluctuation attenuation effect of perforated panels. This work demonstrates the applicability of CFD in gas turbine combustion instability control device design

An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids

In this article we design and analyze a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the linear system of equations stemming from discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polytopic meshes. The preconditioner is based on a coarse space and a non-overlapping partition of the computational domain where local solvers are applied in parallel. In particular, the coarse space can potentially be chosen to be non-embedded with respect to the finer space; indeed it can be obtained from the fine grid by employing agglomeration and edge coarsening techniques. We investigate the dependence of the condition number of the preconditioned system with respect to the diffusion coefficient and the discretization parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: The scalar case

We develop a one--parameter family of hp-version discontinuous Galerkin finite element methods for the numerical solution of quasilinear elliptic equations in divergence-form in a bounded Lipschitz domain. Using Brouwer's Fixed Point Theorem, we show existence and uniqueness of the solution. In addition, we derive an error bound in a broken energy norm which is optimal in h and mildly suboptimal in p

hp-Version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form.

In this paper we consider the a posteriori and a priori analysis of hp-discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. Based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local polynomial-degree variation and local mesh subdivision. The theoretical results are illustrated by a series of numerical experiments

Enhancing SPH using moving least-squares and radial basis functions

In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length -- the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem.Comment: 10 pages, 3 figures, In Proc. A4A5, Chester UK, Jul. 18-22 200

Adaptive Finite Element Simulation of Steady State Currents at Microdisc Electrodes to a Guaranteed Accuracy

We consider the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the "edge-effect". Our approach to overcoming this problem involves the derivation of an a posteriori bound on the error in the numerical approximation for the current which can be used to drive an adaptive mesh-generation algorithm. This allows us to calculate the current to within a prescribed tolerance. Here we demonstrate the power of the method for a simple model problem -- an E reaction mechanism at a microdisc electrode -- for which the analytical solution is known, then we extend the work to the case of a (pseudo) first order EC' reaction mechanism at both an inlaid and a recessed disc

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows

In this article we develop both the a priori and a posteriori error analysis of hp–version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain $\Omega \subset R^{d}, d$ = 2,3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm

Models for pattern formation in somitogenesis: a marriage of cellular and molecular biology

Somitogenesis, the process by which a bilaterally symmetric pattern of cell aggregations is laid down in a cranio-caudal sequence in early vertebrate development, provides an excellent model study for the coupling of interactions at the molecular and cellular level. Here, we review some of the key experimental results and theoretical models related to this process. We extend a recent chemical pre-pattern model based on the cell cycle Journal of Theoretical Biology 207 (2000) 305-316, by including cell movement and show that the resultant model exhibits the correct spatio-temporal dynamics of cell aggregation. We also postulate a model to account for the recently observed spatio-temporal dynamics at the molecular level