H-Adaptive Finite Element Methods for 1D Stationary High Gradient Boundary Value Problems

This article considered the traditional finite element method (FEM) and adaptive finite element method (FEM) for the numerical solution of the one-dimensional boundary value problems. We established the preference or the superiority of the h-adaptive FEM to traditional FEM in high gradient problems in terms of accuracy and cost of computation. Numerical examples which confirm the performance and adaptability of the h-adaptive method over the traditional finite element method and the high accuracy of the numerical solution are presented. Detailed error analysis of linear elements was also discussed. In conclusion, h-adaptive FEM is recommended for complex systems with high gradient problems.

A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering

In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

Diamond-Shaped Extended Fins for Heat Transfer Enhancement in a Double-Pipe Heat Exchanger: An Innovative Design

Heat transfer enhancement in heat exchangers results in thermal efficiency and energy saving. In double-pipe heat exchangers (DPHEs), extended or augmented fins in the annulus of the two concentric pipes, i.e., at the outer surface of the inner pipe, are used to extend the surface of contact for enhancing heat transfer. In this article, an innovative diamond-shaped design of extended fins is proposed for DPHEs. This type of fin is considered for the first time in the design of DPHEs. The triangular-shaped and rectangular-shaped fin designs of DPHE, available in the literature, can be recovered as special cases of the proposed design. An h-adaptive finite element method is employed for the solution of the governing equations. The results are computed for various performance measures against the emerging parameters. The results dictate that the optimal configurations of the diamond-shaped fins in the DPHE for an enhanced heat transfer are recommended as follows: If around 4–6, 8–12, or 16–32 fins are to be placed in the DPHE, then the height of the fins should be 20%, 80%, or 100%, respectively, of the annulus width. If frictional loss of heat is also to be considered, then for fin-heights of 20–80% and 100% of the annulus width, the placement of 4 and 8 diamond-shaped fins, respectively, is recommended for an enhanced heat transfer. These recommendations are for the radii ratio (i.e., the ratio of the inner pipe radius to that of the outer pipe) of 0.25. The recommendations are be modified if the radii ratio is altered.

Time-Adaptive Determination of Drug Efficacy in Mathematical Model of HIV Infection

The paper considers a time-adaptive finite element method for determination of drug efficacy in a parameter identification problem (PIP) for a system of ordinary differential equations (ODE) that describes dynamics of the primary human immunodeficiency virus (HIV) infection with drug therapy. Tikhonov’s regularization method, optimization approach and finite element method to solve this problem are presented. A posteriori error estimates in the Tikhonov’s functional and reconstructed parameter are derived. Based on these estimates a time adaptive algorithm is formulated and numerically tested for different scenarios of noisy observations of virus population function. Numerical results show a significant improvement of reconstruction of drug efficacy parameter using the local time-adaptive mesh refinement method compared to the gradient method applied on a uniform time mesh.

Adaptive Finite Element Method for Solving Poisson Partial Differential Equation

Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.