An element‐wise, locally conservative Galerkin (LCG) method for solving diffusion and convection–diffusion problems

Abstract

SUMMARYAn element-wise locally conservative Galerkin (LCG) method is employed to solve the conservation equations of diffusion and convection-diffusion. This approach allows the system of simultaneous equations to be solved over each element. Thus, the traditional assembly of elemental contributions into a global matrix system is avoided. This simplifies the calculation procedure over the standard global (continuous) Galerkin method, in addition to explicitly establishing element-wise flux conservation. In the LCG method, elements are treated as sub-domains with weakly imposed Neumann boundary conditions. The LCG method obtains a continuous and unique nodal solution from the surrounding element contributions via averaging. It is also shown in this paper that the proposed LCG method is identical to the standard global Galerkin (GG) method, at both steady and unsteady states, for an inside node. Thus, the method, has all the advantages of the standard GG method while explicitly conserving fluxes over each element.Several problems of diffusion and convection-diffusion are solved on both structured and unstructured grids to demonstrate the accuracy and robustness of the LCG method. Both linear and quadratic elements are used in the calculations. For convection-dominated problems, Petrov-Galerkin weighting and highorder characteristic-based temporal schemes have been implemented into the LCG formulation.

A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronary flow using the locally conservative Galerkin (LCG) method

“…The LCG method is a technique introduced by Thomas and Nithiarasu [45][46][47], which treats each element as a sub-domain with its own boundaries. In this case, Equation (51) is expressed over an elemental sub-domain ( e ):…”

“…Information is transmitted between elements via the flux term that is imposed as a Neumann boundary condition. It can be shown that the LCG method is equivalent to the global Galerkin method for convection-diffusion-type problems except on global boundaries [45]. One advantage of LCG method is that only small equations need to be solved; for 1D problems, the 2×2 matrices can be evaluated directly before coding, which removes the need for any matrix inversions.…”

“…Information is transferred between sub-domains via a post-processed boundary flux term. Previous work [45][46][47] has shown that in 1D convectiondiffusion problems, LCG produces identical results to global Galerkin methods, but with the advantage that local conservation is explicitly enforced. The LCG method allows control over individual elements and thus eliminates the need for large matrix inversions, significantly reducing computation time and simplifying code.…”

“…The non-linear equations have been discretized using the finite difference method [2,7,44], as well as various other methods such as the characteristic [16], Galerkin least-squares [6] and discontinuous Galerkin methods [3][4][5]. In this work, we use the recently developed locally conservative Galerkin (LCG) finite element method [45] to solve the governing equations. The LCG method treats each element as an independent sub-domain with weakly imposed Neumann boundary conditions.…”

Application of a locally conservative Galerkin (LCG) method for modelling blood flow through a patient‐specific carotid bifurcation

“…In this section, Equations (4)- (6) are discretized in space using the LCG finite element method [31,33]. As with the global Galerkin form, the variation of each of the variables is approximated by the standard finite element spatial discretization as u i ≈ũ i = Nu i and p ≈p = Np where N are the shape functions.…”