Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface. 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.Comment: 21 pages, 4 figures. IMA Journal of Numerical Analysis 2013, Link to publication: http://imajna.oxfordjournals.org/cgi/content/abstract/drs033? ijkey=45b23qZl5oJslZQ&keytype=re

Analysis of discontinuous Galerkin methods on surfaces

In this thesis, we extend the discontinuous Galerkin framework to surface partial differential equations. This is done by deriving both a priori and a posteriori error estimates for model elliptic problems posed on compact smooth and oriented surfaces in R3, and investigating both theoretical estimates and several generalisations numerically

High order discontinuous Galerkin methods on surfaces

We derive and analyze high order discontinuous Galerkin methods for second-order elliptic problems on implicitely defined surfaces in $\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous Galerkin framework of Arnold et al. [2002] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy norm and the $L^2$ norm.Comment: 23 pages, 2 figure

Adaptive discontinuous Galerkin methods on surfaces

We present a dual weighted residual-based a posteriori error estimate for a discontinuous Galerkin approximation of a surface partial differential equation. We restrict our analysis to a linear second-order elliptic problem posed on hypersurfaces in R3 which are implicitly represented as level sets of smooth functions. We show that the error in the energy norm may be split into a “residual part” and a higher order “geometric part”. Upper and lower bounds for the resulting a posteriori error estimator are proven and we consider a number of challenging test problems to demonstrate the reliability and efficiency of the estimator. We also present a novel “geometric” driven refinement strategy for PDEs on surfaces which considerably improves the performance of the method on complex surfaces

A VALIDATED STABILITY-INDICATING METHOD FOR THE DETERMINATION OF RELATED SUBSTANCES AND ASSAY OF PAMIDRONATE SODIUM PENTAHYDRATE BY HPLC WITHOUT DERIVATIZATION

ABSTRACT A high performance liquid chromatography (HPLC) method is developed for determination related substances and assay in Pamidronate sodium without derivatization. The isocratic method was developed by using column Zorbax SB-C18 (250 mm x 4.6 mm), 5 µm employing the mobile phase containing the buffer (0.4 % v/v n-Hexylamine in water with pH 7.5): Methanol: Acetonitrile (75:25:05 v/v/v). This method is compatible to conductivity detector, Evaporative Light Scattering (ELS) detector and Refractive Index (RI) detector. The Limit of detection (LOD) and limit of quantitation (LOQ) was established for Pamidronate sodium with Conductivity detector, ELS detector and RI detector. The conductivity detector was found to be suitable for related substances determination in Pamidronate sodium when compared with ELS and RI detectors. The LOQ values obtained for phosphoric acid impurity is 0.05% and phosphorus acid impurity is 0.04% with respect to the target analyte concentration (1 mg mL -1 ) by using conductivity detector. The relative response factor was established by linearity method for phosphoric acid impurity and phosphorous acid impurity against Pamidronate sodium by using conductivity detector. The method was validated with respect to accuracy, precision, linearity and robustness by using conductivity detector. Specificity of the method was demonstrated in the presence of known impurities and degradation impurities