Mixed Finite Element Solution Research Articles

On a Stokes hemivariational inequality for incompressible fluid flows with damping

In this paper, a Stokes hemivariational inequality is studied for incompressible fluid flows with the damping effect. The hemivariational inequality feature is caused by the presence of a nonsmooth slip boundary condition of friction type. Well-posedness of the Stokes hemivariational inequality is established through the consideration of a minimization problem. Mixed finite element methods are introduced to solve the Stokes hemivariational inequality and error estimates are derived for the mixed finite element solutions. The error estimates are of optimal order for low-order mixed element pairs under suitable solution regularity assumptions. An efficient iterative algorithm is introduced to solve the mixed finite element system. Numerical results are reported on the performance of the proposed algorithm and the numerical convergence orders of the finite element solutions.

A preserving accuracy two-grid reduced-dimensional Crank-Nicolson mixed finite element method for nonlinear wave equation

In this paper, we mainly resort to a proper orthogonal decomposition (POD) method to study the reduced-dimension of unknown mixed finite element (MFE) solution coefficient vectors in the two-grid Crank-Nicolson MFE (TGCNMFE) method for the nonlinear wave equation and build a preserving accuracy reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method for the nonlinear wave equation. For this purpose, we first build a new TGCNMFE method for the nonlinear wave equation and demonstrate the existence, unconditional stability, and error estimates for the TGCNMFE solutions. From there, we build a preserving accuracy RDETGCNMFE method for the nonlinear wave equation by using the POD method to decrease the dimension of unknown TGCNMFE solution coefficient vectors and employ the matrix analysis to analyze the existence, unconditional stability, convergence, and errors for the RDETGCNMFE solutions. We finally make use of numerical experiments to verify our theoretical results and to show the advantages of RDETGCNMFE method.

The dimension reduction method of two-grid Crank–Nicolson mixed finite element solution coefficient vectors for nonlinear fourth-order reaction diffusion equation with temporal fractional derivative

Herein, we mainly resort to a proper orthogonal decomposition (POD) to study the dimension reduction of unknown solution coefficient vectors in the two-grid Crank–Nicolson mixed finite element (CNMFE) (TGCNMFE) method for the nonlinear fourth-order reaction diffusion equation with temporal fractional derivative and establish a new reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method. For this purpose, we firstly retrospect the TGCNMFE method for the nonlinear fourth-order reaction diffusion equation and provide the existence, unconditional stability, and error estimates of the TGCNMFE solutions. Thereafter, we use the POD method to reduce the dimensionality of the unknown TGCNMFE solution coefficient vectors of the nonlinear fourth-order reaction diffusion equation and develop the new RDETGCNMFE method, and use matrix analysis to analyze the existence, unconditional stability, and errors of the RDETGCNMFE solutions. Finally, we perform numerical experiments to verify our theoretical results and show the advantages of the RDETGCNMFE method. It is worth noting that the RDETGCNMFE method herein is completely distinguished from the existing ones. Hence, the work in this paper is brand-new.

Two-grid dimension reduction method of Crank-Nicolson mixed finite element solution coefficient vectors for the fourth-order extended Fisher-Kolmogorov equation

In this article, we mainly resort to a proper orthogonal decomposition (POD) method to reduce the dimensionality of unknown mixed finite element (MFE) solution coefficient vectors in the two-grid Crank-Nicolson MFE (TGCNMFE) method for the fourth-order extended Fisher-Kolmogorov (FOEFK) equation and to build a new two-grid reduced-dimension extrapolated Crank-Nicolson MFE (TGRDECNMFE) method. For this purpose, we first propose a new TGCNMFE method for the FOEFK equation and demonstrate the existence, unconditional stability, and error estimates of the TGCNMFE solutions. Thereafter, we use the POD method to reduce the dimensionality of the unknown TGCNMFE solution coefficient vectors of the FOEFK equation to develop the new TGRDECNMFE method, and use matrix analysis to analyze the existence, unconditional stability, and errors of TGRDECNMFE solutions. Finally, we perform numerical experiments to confirm our theoretical results and show the advantages of the TGRDECNMFE method.

Two-Grid Method for a Fully Discrete Mixed Finite Element Solution of the Time-Dependent Schrödinger Equation

We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L2-norm with order O(τ+hk+1). Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L2-norm with order O(τ+hk+1+Hk+2). The numerical experiment is provided to demonstrate the efficiency of the algorithm.

Two‐grid mixed finite element method for two‐dimensional time‐dependent Schrödinger equation

In this paper, we study the semidiscrete mixed finite element scheme and construct a two‐grid algorithm for the two‐dimensional time‐dependent Schrödinger equation. We analyze error results of the mixed finite element solution in ‐norm by some projection operators. Then, we propose a two‐grid method of the semidiscrete mixed finite element. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two elliptic equations on the fine grid. We also obtain the error estimate of two‐grid solution with exact solution in ‐norm. Finally, a numerical experiment indicates that our two‐grid algorithm is more efficient than the standard mixed finite element method.

A precision preserving Crank–Nicolson mixed finite element lowering dimension method for the unsteady conduction-convection problem

In this paper, a precision preserving Crank–Nicolson (CN) mixed finite element (MFE) lowering dimension (PPCNMFELD) method for the unsteady nonlinear conduction-convection problem is developed by using proper orthogonal decomposition. The PPCNMFELD method only lowers the dimension of unknown coefficient vectors of MFE solutions for the classical CN MFE (CCNMFE) format but keeps the MFE basis functions unvaried. Under this circumstance, the PPCNMFELD method possesses the same basis functions and the same precision as the CCNMFE method but has fewer unknowns. Thus, the PPCNMFELD method can greatly reduce CPU runtime, slow the accumulation of computing errors in the numerical calculating process, and improve real-time calculation precision. In particular, the existence and stability together with convergence for the PPCNMFELD solutions are proved using a matrix method, which leads to a fine theory and analysis. Additionally, the superiority of the PPCNMFELD method is verified by the numerical experiments for flow around an aerofoil. The PPCNMFELD method possesses time second-order precision and unconditional stability, which are distinct from the other existing reduced-order FE and MFE methods.

Existence and Uniqueness of Generalized and Mixed Finite Element Solutions for Steady Boussinesq Equation

Herein, we mainly employ the fixed point theorem and Lax-Milgram theorem in functional analysis to prove the existence and uniqueness of generalized and mixed finite element (MFE) solutions for two-dimensional steady Boussinesq equation. Thus, we can fill in the gap of research for the steady Boussinesq equation since the existing studies for the equation are assumed the existence and uniqueness of generalized solution without providing proof.

Two-dimensional model for composite beams and its state-space based mixed finite element solution by DQM

Most one-dimensional composite beam theories are based on simplified shear deformation assumptions, or even ignore shear deformation, which have limitations in analyzing multilayer beams with significant material differences. Therefore, this paper firstly proposed a two-dimensional analytical model for composite beams through the equivalent transformation of cross section. Based on the mixed variational principle, the state equations are then derived by finite element meshing and interpolation along the length of the beam, with nodal displacements and their energy-conjugated stresses as state variables. Subsequently, the differential quadrature method (DQM) is introduced to solve the equations and a two-dimensional analysis method for composite beams is established. Due to the use of displacements and stresses as fundamental variables in the state equations, various transfer characteristics of displacements and stresses at the interlayer interfaces of composite beams can be conveniently handled. This method is finally verified by numerical examples of steel-concrete composite beams and concrete beams with corrugated steel webs. Since no assumptions of displacement and stress distributions along the thickness of beams are required, the present method can provide benchmarks for various one-dimensional beam theories.

A Preserving Precision Mixed Finite Element Dimensionality Reduction Method for Unsaturated Flow Problem

The unsaturated flow problem is of important applied background and its mixed finite element (MFE) method can be used to simultaneously calculate both water content and flux in soil, which is the most ideal calculation method. Nonetheless, it includes many unknowns. Thereby, herein we will employ the proper orthogonal decomposition (POD) to lower the dimension of unknown solution coefficient vectors in the MFE method for the unsaturated flow problem. Thus, we first examine the MFE method for the unsaturated flow problem and the existence and convergence of the classical MFE solutions. We then take advantage of the initial L MFE solution coefficient vectors to generate a set of POD basis vectors and utilize the most POD basis vectors to create the preserving precision MFE reduced-dimension (PPMFERD) format. Under the circumstances, the PPMFERD format has the same basis functions as the classical MFE format so that it can maintain the same accuracy as the classical MFE format, but it only includes a few unknowns, so it greatly lightens the calculating load, retards the accumulation of computing errors, saves CPU runtime, and improves the accuracy of the real-time calculation in the computational process. Next, we employ the analysis of matrices to demonstrate the existence and convergence of the PPMFERD solutions such that the theoretical analysis becomes very simple and elegant. Finally, we take advantage of some numerical simulations to check on the correctness of the PPMFERD method. It shows that the PPMFERD method is effective and feasible for simulating both water content and flux in unsaturated flow soil.

The Dimensionality Reduction of Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors for the Unsteady Stokes Equation

By means of a proper orthogonal decomposition (POD) to cut down the dimensionality of unknown finite element (FE) solution coefficient vectors in the Crank–Nicolson (CN) mixed FE (CNMFE) method for two-dimensional (2D) unsteady Stokes equations in regard to vorticity stream functions, a reduced dimension recursive-CNMFE (RDR-CNMFE) method is constructed. In this case, the RDR-CNMFE method has the same FE basis functions and accuracy as the CNMFE method. The existence, stability, and errors of RDR-CNMFE solutions are analyzed by matrix analyzing, resulting in very simple theory analysis. Some numerical tries are used to check on the validity of the RDR-CNMFE method. The RDR-CNMFE method has second-order time accuracy and few unknowns so as to be able to shorten CPU runtime and retard the error cumulation in simulation calculating process, and improve real-time calculating accuracy.

A predictive multiphase model of silica aerogels for building envelope insulations

This work develops a systematic uncertainty quantification framework to assess the reliability of prediction delivered by physics-based material models in the presence of incomplete measurement data and modeling error. The framework consists of global sensitivity analysis, Bayesian inference, and forward propagation of uncertainty through the computational model. The implementation of this framework on a new multiphase model of novel porous silica aerogel materials is demonstrated to predict the thermomechanical performances of a building envelope insulation component. The uncertainty analyses rely on sampling methods, including Markov-chain Monte Carlo and a mixed finite element solution of the multiphase model. Notable features of this work are investigating a new noise model within the Bayesian inversion to prevent biased estimations and characterizing various sources of uncertainty, such as measurements variabilities, model inadequacy in capturing microstructural randomness, and modeling errors incurred by the theoretical model and numerical solutions.

A Reduced-Order Extrapolation (ROE) Method for Solution Coefficient Vectors in the Mixed Finite Element (MFE) Method for the Two-Dimensional (2D) Fourth-Order Hyperbolic Equation

This study focuses on a reduced-order extrapolation method for the coefficient vectors of the mixed finite element solution for two-dimensional fourth-order hyperbolic equation. We first establish the mixed finite element scheme for the equation and give the matrix model of the mixed finite element scheme and the existence, stability and error estimates of its solutions. Then, we derive a reduced-order extrapolation mixed finite element matrix model with a small number of unknowns, where the proper orthogonal decomposition method is used to save central processing unit time, and prove the existence, stability and error estimates of the reduced-order extrapolation mixed finite element solutions with the help of matrix knowledge. More importantly, the reduced-order extrapolation mixed finite element matrix model have the same basis functions and error accuracy as the mixed finite element matrix model. Finally, some numerical experiments confirm the effectiveness of the reduced-order extrapolation mixed finite element matrix model, where the central processing unit time is greatly reduced and the accuracy is maintained.

Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model

In this article, a time two-mesh (TT-M) algorithm combined with the H1-Galerkin mixed finite element (FE) method is introduced to numerically solve the nonlinear distributed-order sub-diffusion model, which is faster than the H1-Galerkin mixed FE method. The Crank-Nicolson scheme with TT-M algorithm is used to discretize the temporal direction at time $t_{n+\frac {1}{2}}$ , the FBN-𝜃 formula is developed to approximate the distributed-order derivative, and the H1-Galerkin mixed FE method is used to approximate the spatial direction. TT-M mixed element algorithm mainly covers three steps: first, the mixed finite element solution of the nonlinear coupled system on the time coarse mesh ΔtC is calculated; next, based on the numerical solution obtained in the first step, the numerical solution of the nonlinear coupled system on time fine mesh ΔtF is obtained by using Lagrange’s interpolation formula; finally, the numerical solution of the linearized system on time fine mesh ΔtF is solved by using the results in the second step. The existence and uniqueness of the solution for our numerical scheme are shown. Moreover, the stability and a priori error estimate are analyzed in detail. Furthermore, numerical examples with smooth and nonsmooth solutions are given to validate our method.

A two‐grid method for characteristic expanded mixed finite element solution of miscible displacement problem

SummaryA combined method consisting of mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics(CEMFEM) for the concentration equation is presented to solve the coupled system of incompressible miscible displacement problem. To solve the resulting nonlinear system of equations efficiently, the two‐grid algorithm relegates all of the Newton‐like iterations to grids much coarser than the original one, with no loss in order of accuracy. It is shown that coarse space can be extremely coarse and our algorithm achieve asymptotically optimal approximation when the mesh sizes satisfy . Numerical experiment is provided to confirm our theoretical results.

Sobolev方程的混合连续时空有限元解的误差估计

The mixed space-time finite element scheme for Sobolev equations was constructed through introduction of auxiliary variables. The scheme can not only reduce the order of the equation with the mixed method, but discretize both space and time variables by means of the finite element technique. A numerical model of high-order accuracy in space and time was obtained. The existence, stability and uniqueness of the mixed space-time finite element solution were proved. The error estimates were derived with the space-time projection operator.

Theoretical stability analysis of mixed finite element model of shale-gas flow with geomechanical effect

In this work, we introduce a theoretical foundation of the stability analysis of the mixed finite element solution to the problem of shale-gas transport in fractured porous media with geomechanical effects. The differential system was solved numerically by the Mixed Finite Element Method (MFEM). The results include seven lemmas and a theorem with rigorous mathematical proofs. The stability analysis presents the boundedness condition of the MFE solution.

A reduced-order extrapolation technique for solution coefficient vectors in the mixed finite element method for the 2D nonlinear Rosenau equation

This study focuses mainly on a reduced-order method for the coefficient vectors in the Crank-Nicolson mixed finite element (CNMFE) solutions for the two-dimensional nonlinear Rosenau equation. We first provide a CNMFE method for the Rosenau equation and give relative theoretical results as well as rewrite the CNMFE method in matrix form. We then produce two sets of proper orthogonal decomposition (POD) bases by means of the first few coefficient vectors of the CNMFE solutions and set up a reduced-order extrapolating CNMFE (ROECNMFE) model with the POD bases as well as discuss the existence and stability as well as errors of the ROECNMFE solutions by means of the matrix tool, resulting in the theoretical analysis becoming very succinct. Lastly, we present some numerical experiments to validate the correctness of the theoretical results, finding that the superiority of the ROECNMFE method is further verified.

Two-grid methods of expanded mixed finite-element solutions for nonlinear parabolic problems

In this paper, we propose two efficient two-grid algorithms for the nonlinear parabolic equations by using expanded mixed finite element methods. To linearize the finite element equations, two-grid algorithms based on some Newton iteration approach and correction method are applied. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h1/3) (or H=O(h1/4)). Experiments are presented to show the efficiency and effectiveness of the two-grid methods.

Analysis of two‐grid discretization scheme for semilinear hyperbolic equations by mixed finite element methods

In this paper, the full discrete scheme of mixed finite element approximation is introduced for semilinear hyperbolic equations. To solve the nonlinear problem efficiently, two two‐grid algorithms are developed and analyzed. In this approach, the nonlinear system is solved on a coarse mesh with width H, and the linear system is solved on a fine mesh with width h≪H. Error estimates and convergence results of two‐grid method are derived in detail. It is shown that if we choose in the first algorithm and in the second algorithm, the two‐grid algorithms can achieve the same accuracy of the mixed finite element solutions. Finally, the numerical examples also show that the two‐grid method is much more efficient than solving the nonlinear mixed finite element system directly.