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We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface in R 3 . An interior penalty (IP) method is introduced on a discrete surface and we derive a-priori error estimates by relating the latter to the original surface via the lift introduced in Dziuk (1988). The estimates suggest that the geometric error terms arising from the surface discretisation do not affect the overall convergence rate of the IP method when using linear ansatz functions. This is then verified numerically for a number of test problems. An intricate issue is the approximation of the surface conormal required in the IP formulation, choices of which are investigated numerically. Furthermore, we present a generic implementation of test problems on surfaces.

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Abstract. We derive and analyze high order discontinuous Galerkin methods for second-order elliptic problems on implicitely defined surfaces in R 3 . This is done by carefully adapting the unified discontinuous Galerkin framework of [3] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy and L 2 norms.Key words. high order discontinuous Galerkin; surface partial differential equations; error analysis.AMS subject classifications. 65N30, 58J05, 65N151. Introduction. Partial differential equations (PDEs) on manifolds have become an active area of research in recent years due to the fact that, in many applications, mathematical models have to be formulated not on a flat Euclidean domain but on a curved surface. For example, they arise naturally in fluid dynamics (e.g., surface active agents on the interface between two fluids, [23]) and material science (e.g., diffusion of species along grain boundaries, [12]) but have also emerged in other areas as image processing and cell biology (e.g., cell motility involving processes on the cell membrane, [27] or phase separation on biomembranes, [21]). Finite element methods (FEMs) for elliptic problems and their error analysis have been successfully applied to problems on surfaces via the intrinsic approach in [17]. This approach has subsequently been extended to parabolic problems [19] as well as evolving surfaces [18]. The literature on the application of FEM to various surface PDEs is now quite extensive, a review of which can be found in [20]. High order error estimates, which require high order surface approximations, have been derived in [15] for the Laplace-Beltrami operator. However, there are a number of situations where conforming FEMs may not be the appropriate numerical method, for instance, problems which lead to steep gradients or even discontinuities in the solution. Such issues can arise for problems posed on surfaces, as in [29] where the authors analyse a model for bacteria/cell aggregation. Without an appropriate stabilisation mechanism artificially added to the surface FEMs scheme, the solution can exhibit a spurious oscillatory behaviour which, in the context of the above problem, leads to negative densities of on-surface living cells. Given the ease with which one can perform hp-adaptivity using high order discontinuous Galerkin (DG) methods and its in-built stabilisation mechanisms for dealing with advection dominated problems and solution blow-ups, it is natural to extend the DG framework for PDEs posed on surfaces. DG methods have first been extended to surfaces in [14], where an interior penalty (IP) method for a linear second-order elliptic problem was introduced and optimal a priori error estimates in the L 2 and energy norms for piecewise linear ansatz functions and surface approximations were derived.

Electrical impedance tomography (EIT) is a noninvasive imaging modality, where imperceptible currents are applied to the skin and the resulting surface voltages are measured. It has the potential to distinguish between ischaemic and haemorrhagic stroke with a portable and inexpensive device. The image reconstruction relies on an accurate forward model of the experimental setup. Because of the relatively small signal in stroke EIT, the finite-element modeling requires meshes of more than 10 million elements. To study the requirements in the forward modeling in EIT and also to reduce the time for experimental image acquisition, it is necessary to reduce the run time of the forward computation. We show the implementation of a parallel forward solver for EIT using the Dune-Fem C++ library and demonstrate its performance on many CPU's of a computer cluster. For a typical EIT application a direct solver was significantly slower and not an alternative to iterative solvers with multigrid preconditioning. With this new solver, we can compute the forward solutions and the Jacobian matrix of a typical EIT application with 30 electrodes on a 15-million element mesh in less than 15 min. This makes it a valuable tool for simulation studies and EIT applications with high precision requirements. It is freely available for download.