Finite Element Spatial Approximation Research Articles

On splitting schemes in the mixed finite element method

Within the research into some geothermal modes, a 3D heat transfer process was described by a first-order system of differential equations (in terms of “temperature-heat-flow”). This system was solved by an explicit scheme for the mixed finite element spatial approximations based on the Raviart-Thomas degrees of freedom. In this paper, several algorithms based on the splitting technique for the vector heat-flow equation are proposed. Some comparison results of accuracy of the algorithms proposed are presented.

A New Approach to Asymptotic Behavior for a Finite Element Approximation in Parabolic Variational Inequalities

The paper deals with the theta time scheme combined with a finite element spatial approximation of parabolic variational inequalities. The parabolic variational inequalities are transformed into noncoercive elliptic variational inequalities. A simple result to time energy behavior is proved, and a new iterative discrete algorithm is proposed to show the existence and uniqueness. Moreover, its convergence is established. Furthermore, a simple proof to asymptotic behavior in uniform norm is given.

Numerical Studies of Discrete Approximations to the Allen–Cahn Equation in the Sharp Interface Limit

The numerical approximations to the Allen-Cahn type diffuse interface models are studied, with a particular focus on their performance in the sharp interface limit and the effectiveness of high order discretization schemes. Different spatial discretizations of an energy functional in the diffuse interface framework are compared first. Discretizations of the time-dependent equation using various different time-stepping schemes and an adaptive finite element spatial approximations are then analyzed. Assessments of the numerical accuracy in different parameter regimes are carried out. The analysis and computational findings provide insight on the effectiveness of the discretization schemes and offer guidance to parameter choices when numerical simulations are conducted within the diffuse interface framework.

An approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method

An approximation scheme is defined for incompressible miscible displacement in porous media. This scheme is constructed by using two methods. Standard mixed finite element is used for the Darcy velocity equation. A characteristics-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest-order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L 2 -norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. This scheme conserves mass globally; in fact, on the discrete level, fluid is transported along the approximate characteristics. Numerical experiments are presented finally to validate the theoretical analysis.

A Posteriori Error Estimates for Parabolic Variational Inequalities

We study a posteriori error estimates in the energy norm for some parabolic obstacle problems discretized with a Euler implicit time scheme combined with a finite element spatial approximation. We discuss the reliability and efficiency of the error indicators, as well as their localization properties. Apart from the obstacle resolution, the error indicators vanish in the so-called full contact set. The case when the obstacle is piecewise affine is studied before the general case. Numerical examples are given.

Locally generated tsunami along the Kaikoura coastal margin: Part 2. Submarine landslides

An examination of the underwater landscape along the northeast coast of the South Island, New Zealand, identified a substantial potential for a submarine landslide in Kaikoura Canyon. A numerical model was applied to calculate runup and inundation arising from a local tsunami generated by such a landslide. The model is based on the Reynolds‐averaged Navier‐Stokes (RANS) equation and used a finite element spatial approximation, implicit time integration, and a semi‐Lagrangian advection approximation. The results indicate that a landslide‐generated tsunami represents a large potential hazard to the area from South Bay to Oaro, South Island, New Zealand, and has the potential to generate large tsunami runup heights along this section of coast. In addition, the tsunami events are characterised by a short time interval between generation and runup.

Locally generated tsunami along the Kaikoura coastal margin: Part 1. Fault ruptures

Tsunami are generated by sudden movements of the ocean bed or by objects such as subaerial landslides and bolides falling into the ocean. An examination of the geophysical setting for the northeast coast of the South Island, New Zealand, identified a substantial potential for submarine fault ruptures and submarine landslides. To examine possible effects of a tsunami, a numerical model was applied to calculate runup and inundation arising from locally‐generated tsunami along this section of the coast. The specific events considered were fault ruptures on the Kekerengu Bank Fault and two lesser faults, and a submarine landslide in Kaikoura Canyon. The model is based on the Reynolds‐averaged Navier‐Stokes (RANS) equation and used a finite element spatial approximation, implicit time integration, a semi‐Lagrangian advection approximation, and a simple method for treating non‐hydrostatic pressure variations. The results indicate that the different generation events have significant effects on different parts of the coastal margin. For the fault‐rupture events, the northern coast of the study area is very exposed to damage from a potential rupture of the Kekerengu Bank Fault. A concern is that the highest waves near Kaikoura would be a result of late arrival of coastally‐trapped waves and would be delayed by up to 1.5h after the fault‐rupture event.

Anisotropic Scattering Treatment for the Neutron Transport Equation with Primal Finite Elements

Three approaches are presented to treat anisotropic scattering in neutron transport. The approaches are based on the even-odd-parity flux formalism and yield three different second-order equations for the even-parity flux. The first one is based on the total elimination of the odd-parity flux of the second-order equation. In the other two approaches, anisotropic scattering contributions are homogenized and incorporated into the collision term. The numerical solutions of these equations are implemented in the CRONOS code for pressurized water reactor core calculations and are done with a finite element spatial approximation and the discrete ordinates methods (S N ) for the angular variable. Numerical results are presented for critical problems (k eff ) in x-y geometry. Comparisons with the APOLLO2 assembly code show the accuracy and the efficiency of the proposed algorithms.

A Characteristics-Mixed Finite Element Method for Advection-Dominated Transport Problems

We define a new finite element method, called the characteristics-mixed method, for approximating the solution to an advection-dominated transport problem. The method is based on a space-time variational form of the advection-diffusion equation. Our test functions are piecewise constant in space, and in time they approximately follow the characteristics of the advective (i.e., hyperbolic) part of the equation. Thus the scheme uses a characteristic approximation to handle advection in time. This is combined with a low-order mixed finite element spatial approximation of the equation. Boundary conditions are incorporated in a natural and mass conservative fashion. The scheme is completely locally conservative; in fact, on the discrete level, fluid is transported along the approximate characteristics. A postprocessing step is included in the scheme in which the approximation to the scalar unknown is improved by utilizing the approximate vector flux. This has the effect of improving the rate of convergence of the method. We show that it is optimally convergent to order one in time and at least suboptimally convergent to order ${3 / 2}$ in space.