Po-Wen Hsieh

A bubble-stabilized least-squares finite element method for steady MHD duct flow problems at high Hartmann numbers

In this paper we devise a stabilized least-squares finite element method using the residual-free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow. We convert the original system of second-order partial differential equations into a first-order system formulation by introducing two additional variables. Then the least-squares finite element method using C^0 linear elements enriched with the residual-free bubble functions for all unknowns is applied to obtain approximations to the first-order system. The most advantageous features of this approach are that the resulting linear system is symmetric and positive definite, and it is capable of resolving high gradients near the layer regions without refining the mesh. Thus, this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the Hartmann number. Before incoorperating the bubble functions into the global problem, we apply the Galerkin least-squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements. Therefore, we indeed introduce a two-level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual-free bubble functions. Numerical results confirming theoretical findings are presented for several examples including the Shercliff problem.

A Novel Least-Squares Finite Element Method Enriched with Residual-Free Bubbles for Solving Convection-Dominated Problems

In this paper we devise a novel least-squares finite element method (LSFEM) for solving scalar convection-dominated convection-diffusion problems. First, we convert a second-order convection-diffusion problem into a first-order system formulation by introducing the gradient

An Unconditionally Energy Stable Penalty Immersed Boundary Method for Simulating the Dynamics of an Inextensible Interface Interacting with a Solid Particle

\mathbf{p}:=-\kappa\nabla u

On Two Iterative Least-Squares Finite Element Schemes for the Incompressible Navier–Stokes Problem

as an additional variable. The LSFEM using continuous piecewise linear elements enriched with residual-free bubbles for both variables

A regularization model with adaptive diffusivity for variational image denoising

u

On efficient least-squares finite element methods for convection-dominated problems

and

A Tailored Finite Point Method for Solving Steady MHD Duct Flow Problems with Boundary Layers

\mathbf{p}

Two new upwind difference schemes for a coupled system of convection-diffusion equations arising from the steady MHD duct flow problems

is applied to solve the first-order mixed problem. The residual-free bubble functions are assumed to strongly satisfy the associated homogeneous second-order convection-diffusion equations in the interior of each element, up to the contribution of the linear part, and vanish on the element boundary. To implement this two-level least-squares approach, a stabilized method of Galerkin/least-squares type is used to approximate the residual-free bubble functions. This enriched LSFEM not only inherits the advantages of the primitive LSFEM, such as the resulting linear system being symmetric and positive definite, but also exhibits the characteristics of the residual-free bubble method without involving stability parameters. Several numerical experiments are given to demonstrate the effectiveness of the proposed enriched LSFEM. The accuracy and computational cost of this enriched LSFEM are also compared with those of the primitive LSFEM. We find that for a small diffusivity

Analysis of a new stabilized finite element method for the reaction–convection–diffusion equations with a large reaction coefficient

\kappa

A new stabilized linear finite element method for solving reaction–convection–diffusion equations

, the enriched LSFEM is much better able to capture the nature of layer structure in the solution than the primitive LSFEM, even if the primitive LSFEM uses a very fine mesh or higher-order elements. In other words, the enriched LSFEM provides a significant improvement, with a lower computational cost, over the primitive LSFEM for solving convection-dominated problems.