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We analyze three discontinuous Galerkin approximations for solving elliptic problems in two or three dimensions. In each one, the basic bilinear form is nonsymmetric: the first one has a penalty term on edges, the second has one constraint per edge, and the third is totally unconstrained. For each of them we prove hp error estimates in the H 1 norm, optimal with respect to h, the mesh size, and nearly optimal with respect to p, the degree of polynomial approximation. We establish these results for general elements in two and three dimensions. For the unconstrained method, we establish a new approximation result valid on simplicial elements. L 2 estimates are also derived for the three methods. Introduction.Over the past five years, discontinuous Galerkin methods have become very widely used for solving a large range of computational fluid problems. They are preferred over more standard continuous methods because of their flexibility in approximating globally rough solutions, their local mass conservation, their possible definition on unstructured meshes, their potential for error control and mesh adaptation, and their straightforward applications to anisotropic materials. Another interesting aspect of these methods is that they can employ polynomials of degree k (P k ) on quadrilaterals, whose dimension is substantially lower than that of Q k , the standard space of functions on quadrilaterals. Similarly, they lead to smaller systems of equations than the locally conservative mixed finite element methods and do not require the solution of saddle-point problems.In this paper, we discuss three numerical algorithms for elliptic problems which employ discontinuous approximation spaces. The three methods are called the nonsymmetric interior penalty Galerkin (NIPG) method, the nonsymmetric constrained Galerkin (NCG) method, and the discontinuous Galerkin (DG) method. The three algorithms are closely related in that the underlying bilinear form for all three is the same and is nonsymmetric.In the NIPG method, we modify the bilinear form of the interior penalty Galerkin method treated by Douglas and Dupont [9], Wheeler [18], Arnold [2], and Wheeler and Darlow [19]. Here, we antisymmetrize the bilinear form by changing the sign of one term and as a consequence we require only a positive penalty, whereas the proofs in [18] assume that the penalty is bounded below by a problem-dependent constant. We choose the penalty term in such a way that we can establish both H 1 and L 2 optimal convergence rates.

Abstract. A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and L 2 estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.

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Inflammation and wound healing are inextricably linked and complex processes, and are deranged in the setting of chronic, nonhealing diabetic foot ulcers (DFU). An ideal therapy for DFU should both suppress excessive inflammation while enhancing healing. We reasoned that biological simulation would clarify mechanisms and help refine therapeutic approaches to DFU. We developed an agent-based model (ABM) capable of reproducing qualitatively much of the literature data on skin wound healing, including changes in relevant cell populations (macrophages, neutrophils, fibroblasts) and their key effector cytokines (tumor necrosis factor-alpha [TNF], interleukin [IL]-1beta, IL-10, and transforming growth factor [TGF]-beta1). In this simulation, a normal healing response results in tissue damage that first increases (due to wound-induced inflammation) and then decreases as the collagen levels increase. Studies by others suggest that diabetes and DFU are characterized by elevated TNF and reduced TGF-beta1, although which of these changes is a cause and which one is an effect is unclear. Accordingly, we simulated the genesis of DFU in two ways, either by (1) increasing the rate of TNF production fourfold or (2) by decreasing the rate of TGF-beta1 production 67% based on prior literature. Both manipulations resulted in increased inflammation (elevated neutrophils, TNF, and tissue damage) and delayed healing (reduced TGF-beta1 and collagen). Our ABM reproduced the therapeutic effect of platelet-derived growth factor/platelet releasate treatment as well as DFU debridement. We next simulated the expected effect of administering (1) a neutralizing anti-TNF antibody, (2) an agent that would increase the activation of endogenous latent TGF-beta1, or (3) latent TGF-beta1 (which has a longer half-life than active TGF-beta1), and found that these therapies would have similar effects regardless of the initial assumption of the derangement that underlies DFU (elevated TNF vs. reduced TGF-beta1). In silico methods may elucidate mechanisms of and suggest therapies for aberrant skin healing.