A Block Finite Difference Scheme for Second-Order Elliptic Problems with Discontinuous Coefficients

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

Real-world groundwater modeling deals with heterogeneous and anisotropic aquifer systems. Numerical models, particularly the finite difference and finite element methods, are extensively used for modeling such a complex aquifer system. The finite difference method is an age-old straightforward approach, whereas the finite element method is more idealistic, relatively new, and has emerged as a popular approach. The unique advantages with the finite element method are (i) an irregular-shaped domain can be discretized into elements with a matching boundary, (ii) a node can be connected with the neighboring nodes in any direction. Thus, the finite element analysis allows water/solute to flow more freely toward the surrounding nodes along nodal links. On the other hand, the mathematical formulation of the finite element technique follows a set of rules step by step. Conventionally, this idealistic approach uses shape functions and integration techniques to develop element matrices, which, in turn, are assembled to form the global matrix. This conventional global matrix was found to be inconsistent with the basic partial differential equation representing the mass balance in a nodal domain as a control volume. As a result, the matrix solution, particularly for nonlinear problems (e.g., vadose zone flow, density dependent flow, and so forth) becomes unstable and computationally burdensome. To overcome the limitations of the conventional finite element method for real-world groundwater modeling, a new approach, called the direct-formulation finite element (DFFE) method, has been developed. In this approach, a two-dimensional model domain is discretized into triangular elements. Element matrices are formed directly based on flow principles and material properties of elements without using shape functions and integration process. The DFFE method is robust. The global matrix is consistent, and the solution is stable and efficient. The results obtained from this approach were found in a good agreement with that obtained from analytical and other widely used models.

A finite element method (FEM) for solution of the conservative full potential flow equation is presented as a means for improving the computation of transonic flows about complex geometries. The method does not require an orthogonal mesh, thus removing a major constraint in grid generation. A standard Galerkin finite element formulation is used in conjunction with the artificial compressibility method which stabilizes the algorithm in transonic flow and allows the capture of embedded shock waves. Solution comparisons, including one test case which stresses the'capabilities of state-of-the-art finite difference methods, verify the FEM accuracy and geometric capability. I. Introduction W HILE the computation of inviscid two- and threedimensional flowfields by finite difference methods has now become commonplace in industry, one of the most important technological deficiencies is in the area of geometric modeling, particularly for transonic flows. The finite element method (FEM) is presented as a way of alleviating some problems of geometric modeling which limit the effectiveness of current finite difference methods. Certain specific advantages of the FEM, if fully exploited, may improve greatly the computational state-of-the-art. Since the governing equations of inviscid transonic potential flow are nonlinear, the method of superposition (i.e., panel methods) is insufficient to yield a proper solution, even though solutions have been obtained by a method combining surface singularities and field differences.1 Also, because of the nonlinearity, similarity laws do not exist which allow subsonic flows to be related to transonic flows. Field methods must be used, typically finite difference or finite element methods, requiring a grid of points to be defined, both on the body surface arid in the flowfield, on which solutions to the governing equations are approximated. Finite difference method (FDM) computer codes generally

Comparison of Eulerian and Lagrangian numerical techniques for the Stokes equations in the presence of strongly varying viscosity

Certain finite difference methods on rectangular grids for second-order elliptic equations are known to yield superconvergent flux approximations. A class of related finite difference methods recently have been defined for triangular meshes by applying special quadrature rules to an extended version of a mixed finite element method by Arbogast, Dawson, and Keenan [Mixed Finite Element Methods as Finite Difference Methods for Solving Elliptic Equations on Triangular Elements, Tech. report 93-53, Dept. of Computational and Applied Mathematics, Rice University, Houston, TX, 1993; in Computational Methods in Water Resources, Kluwer Academic Publishers, Norwell, MA, 1994]; the usual hybrid mixed method can also be applied to meshes of triangular and tetrahedral elements. Unfortunately, the flux vectors from these methods are only first-order accurate. Empirical evidence indicates that a local postprocessing technique described by Keenan in [An Efficient Postprocessor for Velocities from Mixed Methods on Triangular Elements, Dept. of Computational and Applied Mathematics, Tech. report 94-22, Rice University, Houston, TX, 1994] recovers second-order accurate velocities at special points. In this paper, a class of local postprocessing techniques generalizing the one in [An Efficient Postprocessor for Velocities from Mixed Methods on Triangular Elements, Tech. report 94-22, Rice University, Houston, TX, 1994] are presented and analyzed. These postprocessors are shown to recover second-order accurate velocity fields on three lines meshes. Numerical experiments illustrate these results and investigate more general situations, including meshes of tetrahedral elements.

Different numerical methods including 2-D finite element (FE) analysis were implemented for hypothetical raft foundations of high length/breadth ratios to evaluate the reliability of (1) Boussinesq’s method, (2) “rigid-punch” (RP) elastic methods and (3) finite difference (FD) incorporating subgrade reaction theory. A dedicated computer program was developed and used in addition to professional software packages to analyse uniformly loaded rafts of various length/width ratios and stiffnesses, interacting with multi-layered soils of different properties. It was found that the greater the flexibility of the raft the closer was the agreement between the deformations computed from FE and FD methods and the settlements predicted from FE and RP methods. Additionally, the FE method yielded the most realistic predictions principally because of its capability to model and account for the interactions between soil layers as well as the continuity of the soil support. The results revealed that soil stresses computed from elastic methods were inaccurate in cases where a relatively incompressible bedrock existed within depths equivalent to 4 times the raft width or less. This led to a recommendation that in order to predict raft deformations accurately, only those methods that account for both soil-structure and soil–soil interaction should be used in designing raft foundation in practice.

A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative

A coupled water and heat transport mode is established based on the Richards equation to study water flow and heat transport in soil during freezing process. Both the finite difference and finite element method are used in the discretization, respectively. Two different computer programs are written and used to simulate an indoor unidirectional frozen test. The freezing depth, freezing rate and temperature variation are compared among lab tests, finite difference calculation simulation and finite element calculation simulation. Result shows that: the finite difference method has a better performance in freezing depth simulation while the finite element method has a better performance in numerical stability in one-dimensional freezing simulation.

Efficient solver for mixed finite element method and control-volume mixed finite element method in 3-D on Hexahedral grids

In this study, we introduce a novel non-standard finite difference (NSFD) scheme designed to address the challenges posed by stiff initial value problems. Stiffness in differential equations often leads to numerical instability and requires specialized methods for stable and accurate solutions. A novel set of numerical schemes for solving stiff ordinary differential equations caused by the decay of radioactive substances developed. This paper demonstrates the power of normalization in the discretization function. We employed non-local approximation and renormalization of the denominator function to create qualitatively stable schemes for a stiff ordinary differential equation. The schemes' stability properties were verified using numerical experiments. The schemes' performance is evaluated in comparison to other typical finite difference schemes

Numerical approximation of head and flux covariances in three dimensions using mixed finite elements

Bellows compensators are critical components in pipeline systems, designed to absorb thermal expansions, vibrations, and pressure reflections. Ensuring their operational reliability requires accurate prediction of the stress–strain state (SSS) and stability under internal pressure. This study presents a comprehensive mathematical model for analyzing corrugated bellows compensators, formulated as a boundary value problem for a system of partial differential equations (PDEs) within the Kirchhoff–Love shell theory framework. Two numerical approaches are developed and compared: a finite difference method (FDM) applied to a reduced axisymmetric formulation to ordinary differential equations (ODEs) and a finite element method (FEM) for the full variational formulation. The FDM scheme utilizes a second-order implicit symmetric approximation, ensuring stability and efficiency for axisymmetric geometries. The FEM model, implemented in Ansys 2020 R2, provides high fidelity for complex geometries and boundary conditions. Convergence analysis confirms second-order spatial accuracy for both methods. Numerical experiments determine critical pressures based on the von Mises yield criterion and linearized buckling analysis, revealing the influence of geometric parameters (wall thickness, number of convolutions) on failure mechanisms. The results demonstrate that local buckling can occur at lower pressures than that of global buckling for thin-walled bellows with multiple convolutions, which is critical for structural reliability assessment. The proposed combined approach (FDM for rapid preliminary design and FEM for final verification) offers a robust and efficient methodology for bellows design, enhancing reliability and reducing development time. The work highlights the importance of integrating rigorous PDE-based modeling with modern numerical techniques for solving complex engineering problems with a focus on structural integrity and long-term performance.

The numerical solution of the behaviour of discrete time steps in digital computer analysis of square aquifers containing pumped wells is examined by using the finite element method with a 4 node linear quadrilateral isoparametric surface element. A wide range of time steps are used in the computation. The calculations show that discrete time steps can cause errors and oscillations in the calculations particularly when wells start and stop pumping. Comparison with known results obtained by theoretical and finite difference procedures has been considered. The main objective of this paper is to demonstrate comparison of the finite element and finite difference simulation results over a regular linear 4 node quadrilateral mesh suitable to represent the two numerical schemes with a marked similarity. The dimensionless time drawdown results of the finite element method agreed well with the finite difference and analytical results for small time increment. However, for large time increments, there are from slight to significant oscillations in the results and notable discrepancies are observed in the solutions of the two numerical methods.

The computational complexity of the finite difference (FD) schemes for the solution of the plate equation prevents them from being used in musical applications. The explicit FD schemes can be parallelized to run on multi-processor arrays for achieving real-time performance. Field Programmable Gate Arrays (FPGAs) provide an ideal platform for implementing these architectures with the advantages of low-power and small form factor. The paper presents a design for implementing FD schemes for the plate equation on a multi-processor architecture on a FPGA device. The results show that 64 processing elements can be accommodated on a Xilinx X2VP50 device, achieving 487 kHz throughput for a square FD grid of 50×50 points.

The performance of electrical machines is largely dictated by the action of current and flux in the core length. The field in a cross-section obeys Poisson's equation and approximate solutions have been obtained by finite difference and element methods. The finite difference method requires a large number of nodes and is slow to converge as permeability is variable. The finite element method is more flexible being more readily fitted to iron-air boundaries and has better convergence. However, it is difficult to formulate a legitimate variational formulation for transient conditions in the presence of dissipation. Here, discrete equations are formed by applying Ampere's circuital law around each node. Careful choice of contour lines give a current distribution superior to that obtained with finite elements. Fast convergence is obtained and the method is applicable under transient conditions.