Rectangular Finite Element Method Research Articles

High accuracy analysis of a nonconforming rectangular finite element method for the Brinkman model

High accuracy analysis of a nonconforming rectangular finite element method for the Brinkman model

A Uniformly Stable Nonconforming FEM Based on Weighted Interior Penalties for Darcy-Stokes-Brinkman Equations

AbstractA nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

The Highest Superconvergence of the Tri-linear Element for Schr $$\ddot{\text {o}}$$ o ¨ dinger Operator with Singularity

In this paper, the eigenvalues for Schr$$\ddot{\text {o}}$$o¨dinger operator with singularity are analyzed. A special piecewise uniform rectangular partition is constructed and it has been proven that, under this partition, the tri-linear rectangular finite element method has the highest possible superconvergence rate for eigenvalue.

Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems

AbstractIn this article, we consider rectangular finite element methods for fourth order elliptic singular perturbation problems. We show that the non‐ C0 rectangular Morley element is uniformly convergent in the energy norm with respect to the perturbation parameter. We also propose a C0 extended high order rectangular Morley element and prove the uniform convergence. Finally, we do some numerical experiments to confirm the theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

A low order anisotropic nonconforming characteristic finite element method for a convection-dominated transport problem

A low order anisotropic nonconforming rectangular finite element method for the convection–diffusion problem with a modified characteristic finite element scheme is studied in this paper. The O ( h 2 ) order error estimate in L 2 -norm with respect to the space, one order higher than the expanded characteristic-mixed finite element scheme with order O ( h ) , and the same as the conforming case for a modified characteristic finite element scheme under regular meshes, is obtained by use of some distinct properties of the interpolation operator and the mean value technique, instead of the so-called elliptic projection, which is an indispensable tool in the convergence analysis of the previous literature. Lastly, some numerical results of the element are provided to verify our theoretical analysis.

Superconvergence of rectangular finite element with interpolated coefficients for semilinear elliptic problem

In this paper, the rectangular finite element method with interpolated coefficients for the second-order semilinear elliptic problem is introduced and analyzed. Assume that Ω is polygonal domain, and rectangular partition is quasiuniform and denote by Z 0 and Z 1 the sets of ( n + 1)-order Lobatto points and n-order Gauss points of all elements, respectively. Based on an orthogonal expansion in an element, and on the property of corresponding finite element for an auxiliary linear elliptic problem, optimal superconvergence u − u h = O( h n+2 ), n ⩾ 2, at z ∈ Z 0 and D( u − u h ) = O( h n+1 ), n ⩾ 1, at z ∈ Z 1 are proved, respectively. It is shown that the finite element with interpolated coefficients has the same superconvergence as that of classical finite elements, but more economic and efficient. Finally the results in the case of quadratic finite element are supported by a numerical example.