A finite element formulation for a convection–diffusion equation based on Cattaneo’s law

2007 ◽  
Vol 196 (9-12) ◽  
pp. 1757-1766 ◽  
Author(s):  
Héctor Gómez ◽  
Ignasi Colominas ◽  
Fermín Navarrina ◽  
Manuel Casteleiro
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Cattaneo’S Law

Hybrid Nodal Integral / Finite Element Method for Time-Dependent Convection Diffusion Equation

2021 ◽  
Author(s):  
Sundar Namala ◽  
Rizwan Uddin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Numerical Solutions ◽  
Second Order ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Integral Methods ◽  
Element Method

Abstract Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). The standard application of NIM is restricted to domains that have boundaries parallel to one of the coordinate axes/palnes (in 2D/3D). The hybrid nodal-integral/finite-element method (NI-FEM) reported here has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM, and the rest of the domain can be discretized and solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM regions and FEM regions. We here report the development of hybrid NI-FEM for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Resulting hybrid numerical scheme is implemented in a parallel framework in Fortran and solved using PETSc. The preliminary approach to domain decomposition is also discussed. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is more efficient compared to standalone conventional numerical schemes like FEM.

The Inverse Least-Squares Finite Element Method Applied to the Convection-Diffusion Equation

2009 ◽  
Author(s):  
Brian H. Dennis
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Boundary Conditions ◽  
Least Squares ◽  
Heat Fluxes ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Current Formulation ◽  
Element Method

A Least Squares Finite Element Method (LSFEM) formulation for the detection of unknown boundary conditions in problems governed by the steady convection-diffusion equation will be presented. The method is capable of determining temperatures, and heat fluxes in location where such quantities are unknown provided such quantities are sufficiently over-specified in other locations. For the current formulation it is assumed the velocity field is known. The current formulation is unique in that it results in a sparse square system of equations even for partial differential equations that are not self-adjoint. Since this formulation always results in a symmetric positive-definite matrix, the solution can be found with standard sparse matrix solvers such as preconditioned conjugate gradient method. In addition, the formulation allows for equal order approximation of temperature and heat fluxes as it is not subject to the inf-sup condition. The formulation allow for a treatment of over-specified boundary conditions. Also, various forms of regularization can be naturally introduced within the formulation. Details of the discretization and sample results will be presented.

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