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The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors’ previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems.

We here employ a proper orthogonal decomposition (POD) to reduce the dimensionality of unknown coefficient vectors of finite element (FE) solutions for the fractional Tricomi-type equation and develop a reduced-dimension extrapolating FE (RDEFE) method for the fractional Tricomi-type equation. For this purpose, we first develop an FE method for the fractional Tricomi-type equation and provide the existence, unconditional stability, and error analysis for the FE solutions. We then develop the RDEFE method for the fractional Tricomi-type equation by means of the POD technique and analyze the existence, unconditional stability, and errors for the RDEFE solutions by using the matrix analysis. Lastly, we provide two numerical examples to verify our theoretical results and to explain the advantages of the RDEFE method.

Abstract We develop the Crank–Nicolson finite element (CNFE) method for the two-dimensional (2D) uniform transmission line equation, study the stability and existence as well as error estimates for the CNFE solutions of the 2D uniform transmission line equation by strict theoretical approaches. We verify the correctness of the obtained theoretical results by means of numerical tests.

Abstract In this article, we mainly develop a reduced order extrapolating model for the solution coefficient vectors of the classical collocation spectral (CCS) scheme to the two-dimensional (2D) telegraph equation by means of a proper orthogonal decomposition (POD). Therefore, we first present the CCS scheme, offer the existence, stability, and error estimates to the SC solutions, and rewrite the CCS scheme into a matrix-form. We then build a reduced order extrapolating collocation spectral (ROECS) model and analyze the existence and stability as well as errors of the ROECS solutions by some matrix tools. We finally verify the reliability and validity of the ROECS model by means of two sets of numerical simulations for the magnetic field produced by two parallel wires with the same voltage.

Abstract In this study, a time semi-discretized Crank–Nicolson (CN) scheme of the two-dimensional (2D) unsteady conduction–convection problems for vorticity and stream functions is first built together with showing the existence and stability along with error estimates to the semi-discretized CN solutions. Afterwards, a fully discretized spectral element CN (SECN) model of the 2D unsteady conduction–convection problems as regards the vorticity and stream functions is set up together with showing the proof of the existence and stability along with error estimates of the SECN solution. Lastly, a set of numerical experiments are offered for checking the correctness of the theoretical conclusions.

The mixed finite element (MFE) method is one of the most valid numerical approaches to solve hydrodynamic equations because it can be very suited to solving problems with complex computing domains. Regrettably, the MFE method for the hydrodynamic equations would include lots of unknowns. Especially, when it is applied to settling the practical engineering problems, it could contain hundreds of thousands and even tens of millions of unknowns. Thus, it would bring about many difficulties for actual applications, such as consuming a long CPU running time and accumulating many round-off errors, so as to be very difficult to obtain the desired numerical solutions. Therefore, we herein take the two-dimensional (2D) unsteady Navier–Stokes equation in hydrodynamics as an example. Using the proper orthogonal decomposition to lower the dimension of unknown Crank–Nicolson MFE (CNMFE) solution coefficient vectors for the 2D unsteady Navier–Stokes equation about vorticity–stream functions, we construct a reduced-dimension recursive CNMFE (RDRCNMFE) method with unchanged basis functions. In the circumstances, the RDRCNMFE method can keep the basis functions unchanged in an MFE subspace and has the same precision as the classical CNMFE method. We employ the matrix method to analyse the existence and stability along with errors to the RDRCNMFE solutions, leading to a very simple theory analysis. We use the numerical simulations for the backwards-facing step flow to verify the effectiveness of the RDRCNMFE method. The RDRCNMFE method with unchanged basis functions only reduces the dimension of the solution coefficient vectors of the CNMFE, which is completely different from previous order reduction methods which greatly affects the accuracy by reducing the dimension of the MFE subspace.

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.

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.

We herein mainly employ a proper orthogonal decomposition (POD) to study the reduced dimension of unknown solution coefficient vectors in the Crank–Nicolson finite element (FE) (CNFE) method for the symmetric tempered fractional diffusion equation so that we can build the reduced-dimension recursive CNFE (RDRCNFE) method. In this case, the RDRCNFE method keeps the same basic functions and accuracy as the CNFE method. Especially, we adopt the matrix analysis to discuss the stability and convergence of RDRCNFE solutions, resulting in the very laconic theoretical analysis. We also use some numerical simulations to confirm the correctness of theoretical results.

Abstract In this paper, by means of a proper orthogonal decomposition (POD) we mainly reduce the order of the classical Crank–Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs). Toward this end, we will first review the CCNFD model for FOPTSGEs and the theoretical results (such as existence, stabilization, and convergence) of the CCNFD solutions. Then we establish an optimized Crank–Nicolson finite difference extrapolating (OCNFDE) model, including very few unknowns but holding the fully second-order accuracy for FOPTSGEs via POD. Next, by a matrix analysis we will discuss the existence, stabilization, and convergence of the OCNFDE solutions. Finally, we will use a numerical example to validate the validity of theoretical conclusions. Moreover, we show that the OCNFDE model is very valid for settling FOPTSGEs.

In this study, we mainly employ a proper orthogonal decomposition (POD) to lower the dimension for the unknown Crank–Nicolson finite element (FE) (CNFE) solution coefficient vectors of the viscoelastic wave (VW) equation so as to build a reduced-dimension recursive CNFE (RDRCNFE) algorithm, adopt matrix analysis to analyze the stability together with errors to the RDRCNFE solutions, and utilize some numerical experimentations to verify the effectiveness of the RDRCNFE algorithm.

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.

Abstract In this article, we first develop a semi-discretized Crank–Nicolson format about time for the two-dimensional non-stationary Stokes equations about vorticity–stream functions and analyze the existence, uniqueness, stability, and convergence of the semi-discretized Crank–Nicolson solutions. Then we establish a fully discretized Crank–Nicolson finite spectral element format based on the quadrilateral elements for the two-dimensional non-stationary Stokes equations about vorticity–stream functions and analyze the existence, uniqueness, stability, and convergence of the Crank–Nicolson finite spectral element solutions. In the end, we use three numerical examples to confirm the validity of our theoretical conclusions.

Abstract In this paper, we mainly focus to study the Crank–Nicolson collocation spectral method for two-dimensional (2D) telegraph equations. For this purpose, we first establish a Crank–Nicolson collocation spectral model based on the Chebyshev polynomials for the 2D telegraph equations. We then discuss the existence, uniqueness, stability, and convergence of the Crank–Nicolson collocation spectral numerical solutions. Finally, we use two sets of numerical examples to verify the validity of theoretical analysis. This implies that the Crank–Nicolson collocation spectral model is very effective for solving the 2D telegraph equations.

Abstract In this paper, we focus on the study of the Crank–Nicolson finite difference (CNFD) scheme for the Riesz space fractional-order parabolic-type sine-Gordon equation (RSFOPTSGE). For this purpose, we first establish the CNFD scheme for RSFOPTSGE. Then, we discuss the existence, uniqueness, stability, and convergence of the CNFD solutions. Finally, we supply a numerical experiment to validate the correctness of theoretical results. This indicates that the CNFD scheme is very effective for solving RSFOPTSGE.

Abstract In this study, we first present a classical finite element (FE) method for a two-dimensional (2D) viscoelastic wave equation and analyze the existence, stability, and convergence of the FE solutions. Then we establish an optimized FE extrapolating (OFEE) method based on a proper orthogonal decomposition (POD) method for the 2D viscoelastic wave equation and analyze the existence, stability, and convergence of the OFEE solutions and furnish the implement procedure of the OFEE method. Finally, we furnish a numerical example to verify that the numerical computing results correspond with the theoretical ones. This signifies that the OFEE method is feasible and efficient for solving the 2D viscoelastic wave equation.

Abstract In this paper, we devote ourselves to establishing the unconditionally stable and absolutely convergent optimized finite difference Crank-Nicolson iterative (OFDCNI) scheme containing very few degrees of freedom but holding sufficiently high accuracy for the two-dimensional (2D) Sobolev equation by means of the proper orthogonal decomposition (POD) technique, analyzing the stability and convergence of the OFDCNI solutions and using the numerical simulations to verify the feasibility and effectiveness of the OFDCNI scheme.

Abstract In this study, we devote ourselves to establishing a stabilized mixed finite element (MFE) reduced-order extrapolation (SMFEROE) model holding seldom unknowns for the two-dimensional (2D) unsteady conduction-convection problem via the proper orthogonal decomposition (POD) technique, analyzing the existence and uniqueness and the stability as well as the convergence of the SMFEROE solutions and validating the correctness and dependability of the SMFEROE model by means of numerical simulations.

Abstract In this article, a functional minimum problem equivalent to the p-Laplace equation is introduced, a finite element-Newton iteration formula is established, and a well-posed condition of iterative functions satisfied is provided. According to the well-posed condition, an effective initial iterative function is presented. Using the effective particular initial function and Newton iterations with the iterative step length equal to 1, an effective particular sequence of iterative functions is obtained. With the decreasing properties of gradient modulus of subdivision finite element, it has been proved that the function sequence converges to the solution of finite element formulation of p-Laplace equation. Moreover, a discussion on local convergence rate of iterative functions is provided. In summary, the iterative method based on the effective particular initial function not only makes up the shortage of the Newton algorithm, which requires an exploratory reduction in the iterative step length, but also retains the benefit of fast convergence rate, which is verified with theoretical analysis and numerical experiments.

Abstract In this study, a highly efficient spectral-Galerkin method is posed for the fourth-order Steklov equation with boundary eigenvalue. By making use of the spectral theory of compact operators and the error formulas of projective operators, we first obtain the error estimates of approximative eigenvalues and eigenfunctions. Then we build a suitable set of basis functions included in H 0 1 ( Ω ) ∩ H 2 ( Ω ) $H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$ and establish the matrix model for the discrete spectral-Galerkin scheme by adopting the tensor product. Finally, we use some numerical experiments to verify the correctness of the theoretical results.

Abstract In this paper, we provide a new type of study approach for the two-dimensional (2D) Sobolev equations. We first establish a semi-discrete Crank-Nicolson (CN) formulation with second-order accuracy about time for the 2D Sobolev equations. Then we directly establish a fully discrete CN finite volume element (CNFVE) formulation from the semi-discrete CN formulation about time and provide the error estimates for the fully discrete CNFVE solutions. Finally, we provide a numerical example to verify the correction of theoretical conclusions. Further, it is shown that the fully discrete CNFVE formulation is better than the fully discrete FVE formulation with first-order accuracy in time.

Abstract In this study, a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on a spherical domain is developed. By employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, the error estimates of approximate eigenvalues and eigenfunctions are provided. By adopting orthogonal spherical base functions, the discrete model with sparse mass and stiff matrices is established so that it is very efficient for finding the numerical solutions of biharmonic eigenvalue equations on the spherical domain. Some numerical examples are provided to validate the theoretical results.

This study is mainly concerned with the reduced-order extrapolating technique about the unknown solution coefficient vectors in the Crank-Nicolson finite element (CNFE) method for the parabolic type partial differential equation (PDE). For this purpose, the CNFE method and the existence, stability, and error estimates about the CNFE solutions for the parabolic type PDE are first derived. Next, a reduced-order extrapolating CNFE (ROECNFE) model in matrix-form is established with a proper orthogonal decomposition (POD) method, and the existence, stability, and error estimates of the ROECNFE solutions are proved by matrix theory, resulting in an graceful theoretical development. Specially, our study exposes that the ROECNFE method has the same basis functions and the same accuracy as the CNFE method. Lastly, some numeric tests are shown to computationally verify the validity and correctness about the ROECNFE method.

In this study, a reduced-order extrapolation spectral-finite difference (ROESFD) scheme based on the proper orthogonal decomposition (POD) method is set up for the two-dimensional (2D) second-order hyperbolic equations. First, the classical spectral-finite difference (CSFD) method for the 2D second-order hyperbolic equations and its stability, convergence, and flaw are introduced. Then, a new ROESFD scheme that has very few degrees of freedom but holds sufficiently high accuracy is set up by the POD method and its implementation is offered. Finally, three numerical examples are offered to explain the validity of the theoretical conclusion. This implies that the ROESFD scheme is viable and efficient for searching the numerical solutions of the 2D second-order hyperbolic equations.

We firstly employ a proper orthogonal decomposition (POD) method, Crank–Nicolson (CN) technique, and two local Gaussian integrals to establish a PODbased reduced-order stabilized CN mixed finite element (SCNMFE) formulation with very few degrees of freedom for non-stationary parabolized Navier–Stokes equations. Then, the error estimates of the reduced-order SCNMFE solutions, which are acted as a suggestion for choosing number of POD basis and a criterion for updating POD basis, and the algorithm implementation for the POD-based reduced-order SCNMFE formulation are provided, respectively. Finally, some numerical experiments are presented to illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order SCNMFE formulation is feasible and efficient for finding numerical solutions of the non-stationary parabolized Navier–Stokes equations.