Accurate FIC-FEM formulation for the multidimensional steady-state advection–diffusion–absorption equation

Abstract

In this paper we present an accurate stabilized FIC-FEM formulation for the multidimensional steady-state advection-diffusionabsorption equation. The stabilized formulation is based on the Galerkin FEM solution of the governing differential equations derived via the Finite Increment Calculus (FIC) method using two stabilization parameters. The value of the two stabilization parameters ensuring an accurate nodal FEM solution using uniform meshes of linear elements is obtained from the optimal values for the 1D problem. The accuracy of the new FIC-FEM formulation is demonstrated in the solution of 2D steady-state advection-diffusionabsorption problems for a range of physical parameters and boundary conditions. c

Fast explicit time integration schemes for parabolic problems in mechanics

“…The FIC approach for formulating higher order forms of the differential equation in mechanics has been successfully used, in conjunction with standard numerical methods, such as the FEM and meshless procedures, for finding accurate and stable solutions for steady state and transient problems, such as convection-diffusion-radiation, fluid dynamics and quasi-incompressible solids, among others. 6,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] A general FIC-Time form of the semidiscrete parabolic equation can be written as follows…”

A Review on Thermo-mechanical Modelling of Arch Dams During Construction and Operation: Effect of the Reference Temperature on the Stress Field

“…with D ijkl being the fourth-order constitutive tensor of the material, ε kl the deformation tensor, β ij the thermal expansion tensor, and T ref the reference temperature of the structure. Different stabilisation techniques can be found in the literature to solve the heat transfer problem without spurious oscillations [32,81]. In this work, we modified eq.…”

Semi-Lagrangian formulation for the advection–diffusion–absorption equation

“…For absorptiondominated cases, Gibbs oscillations can be found near the Dirichlet boundaries and in regions where the distributed source term is nonregular. Also, the solution of the transient problem may exhibit dispersive oscillations when the initial solution and/or the distributed source term are nonregular [65]. Various techniques for solving these problems can be found in literature, such as the Petrov-Galerkin method [5,23,24,31,34], the Galerkin Least Squares (GLS) method [17,25], the Variational Multiscale (VMS) method [26] or the characteristic split procedure [76,78].…”

“…The accuracy of the Petrov-Galerkin method can however be improved with the FIC stabilization technique [64,66,65]. In 2000 and 2006, respectively, Young et al studied several Eulerian-Lagrangian methods such as the Eulerian-Lagrangian Boundary Element Method [74], which provided the solution for low numerical diffusion, and the Eulerian-Lagrangian method of fundamental solutions [75], which is a mesh-free method that has the simplicity of a fixed grid from the Eulerian method and the computational power of the Lagrangian method.…”