An operator-splitting heterogeneous finite element method for population balance equations: Stability and convergence

We present a heterogeneous finite element approximation of the solution of a population balance equation, which depends both the physical and internal property coordinates. We employ the operator-splitting method to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. It is demonstrated that the variational form of the operator-split population balance equation is equivalent to the variational form of the standard equation up to a perturbation term of order $au^2$ in the backward Euler scheme, where $au$ is a time step. Further, the stability and error estimates have been derived for the heterogeneous finite element discretization scheme applied to the population balance equation. It is shown that a slightly more regularity, $i.e,$ the mixed partial derivatives of the solution has to be bounded, is necessary for the solution of the population balance equation with the operator-splitting finite element method. Numerical results are presented to demonstrate the accuracy of the numerical scheme

An operator-splitting heterogeneous finite element method for population balance equations: stability and convergence

We present a heterogeneous finite element approximation of the solution of a population balance equation, which depends both the physical and internal property coordinates. We employ the operator-splitting method to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. It is demonstrated that the variational form of the operator-split population balance equation is equivalent to the variational form of the standard equation up to a perturbation term of order $tau^2$ in the backward Euler scheme, where $tau$ is a time step. Further, the stability and error estimates have been derived for the heterogeneous finite element discretization scheme applied to the population balance equation. It is shown that a slightly more regularity, $i.e,$ the mixed partial derivatives of the solution has to be bounded, is necessary for the solution of the population balance equation with the operator-splitting finite element method. Numerical results are presented to demonstrate the accuracy of the numerical scheme

Generalized local projection stabilized nonconforming finite element methods for Darcy equations

An a priori analysis for a generalized local projection stabilized finite element solution of the Darcy equations is presented in this paper. A first-order nonconforming P nc 1 finite element space is used to approximate the velocity, whereas the pressure is approximated using two different finite elements, namely piecewise constant P0 and piecewise linear nonconforming P nc 1 elements. The considered finite element pairs, P nc 1 /P0 and P nc 1 /P nc 1 , are inconsistent and incompatibility, respectively, for the Darcy problem. The stabilized discrete bilinear form satisfies an inf-sup condition with a generalized local projection norm. Moreover, a priori error estimates are established for both finite element pairs. Finally, the validation of the proposed stabilization scheme is demonstrated with appropriate numerical examples

AI-augmented stabilized finite element method

An artificial intelligence-augmented Streamline Upwind/Petrov-Galerkin finite element scheme (AiStab-FEM) is proposed for solving singularly perturbed partial differential equations. In particular, an artificial neural network framework is proposed to predict optimal values for the stabilization parameter. The neural network is trained by minimizing a physics-informed cost function, where the equation's mesh and physical parameters are used as input features. Further, the predicted stabilization parameter is normalized with the gradient of the Galerkin solution to treat the boundary/interior layer region adequately. The proposed approach suppresses the undershoots and overshoots in the stabilized finite element solution and outperforms the existing neural network-based partial differential equation solvers such as Physics-Informed Neural Networks and Variational Neural Networks.Comment: 23 pages, 5 figures and 8 table

An overlapping local projection stabilization for Galerkin approximations of Stokes and Darcy flow problems

An a priori analysis for a generalized local projection stabilized finite element approximation of the Stokes, and the Darcy flow equations are presented in this paper. A first-order conforming PC1 finite element space is used to approximate both the velocity and pressure. It is shown that the stabilized discrete bilinear form satisfies the inf-sup condition in the generalized local projection norm. Moreover, a priori error estimates are established in a mesh-dependent norm as well as in the L2 -norm for the velocity and pressure. The optimal and quasi-optimal convergence properties are derived for the Stokes and the Darcy flow problems. Finally, the derived estimates are numerically validated with appropriate examples