A reduced-order energy-stability-preserving finite difference iterative scheme based on POD for the Allen-Cahn equation

A reduced-order extrapolated finite difference iterative method for the Riemann-Liouville tempered fractional derivative equation

The reduced-dimension technique for the unknown solution coefficient vectors in the Crank–Nicolson finite element method for the Sobolev 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.

/pdf/the-dimensionality-reduction-of-crank-nicolson-mixed-finite-1fa0pjuv.pdf

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

In this article, we primarily focuses to study the order‐reduction for the classical natural boundary element (NBE) method for the two‐dimensional (2D) hyperbolic equation in unbounded domain. To this end, we first build a semi‐discretized format about time for the hyperbolic equation and discuss the existence, stability, and convergence of the time semi‐discretized solutions. We then establish the classical fully discretized NBE format from the time semi‐discretized one and analyze the existence, stability, and convergence of the classical NBE solutions. Next, using proper orthogonal decomposition method, we build a reduced‐order extrapolated NBE (ROENBE) format containing very few unknowns but having adequately high accuracy, and we also discuss the existence, stability, and convergence of the ROENBE solutions. Finally, we use some numerical examples to show that the ROENBE method is far superior to the classical NBE one. It shows that the ROENBE method is reliable and effective for solving the 2D hyperbolic equation with the unbounded domain.

A reduced-order extrapolated natural boundary element method based on POD for the 2D hyperbolic equation in unbounded domain

In this article, we devote ourselves to the research of order reduction of natural boundary element (NBE) based on a proper orthogonal decomposition (POD) for the parabolic equation in the two-dimensional (2D) unbounded domain. For this purpose, we first build a NBE format for the parabolic equation in the 2D unbounded domain and discuss the existence, stability, and convergence of the NBE solutions. And then, we build a reduced-order NBE extrapolated (RONBEE) format based on POD, analyze the errors between the classical NBE and RONBEE solutions, and supply the implementation procedure for the RONBEE format. Finally, we utilize some numerical experiments to validate that the numerical computational consequences are consistent with the theoretical ones such that the effectiveness and feasibility of the RONBEE format are further verified.

A reduced-order extrapolated natural boundary element method based on POD for the parabolic equation in the 2D unbounded domain

In this article, we devote ourselves to establishing a natural boundary element (NBE) method for the Sobolev equation in the 2D unbounded domain. To this end, we first constitute the time semi-discretized super-convergence format for the Sobolev equation by means of the Newmark method. Then, using the principle of natural boundary reduction, we establish a fully discretized NBE format based on the natural integral equation and the Poisson integral formula of this problem and analyze the errors between the exact solution and the fully discretized NBE solutions. Finally, we use some numerical experiments to verify that the NBE method is effective and feasible for solving the Sobolev equation in the 2D unbounded domain.

/pdf/a-natural-boundary-element-method-for-the-sobolev-equation-27bcc1p751.pdf

A natural boundary element method for the Sobolev equation in the 2D unbounded domain

This note provides an error estimate for finite element approximation to elliptic partial differential equations (PDEs) with discontinuous Dirichlet boundary data. Solutions of problems of this type are not in H1 and, hence, the standard variational formulation is not valid. To circumvent this difficulty, an error estimate of a finite element approximation in the W1,r(Ω) (0<r<2) norm is obtained through a regularization by constructing a continuous approximation of the Dirichlet boundary data. 

An Error Estimate For Finite Element Approximation to Elliptic PDEs With Discontinuous Dirichlet Boundary Data

In this paper, we present a class of discontinuous Galerkin finite element methods for advection-diffusion-reaction problems and establish a priori error estimates when the solution is only in H1+s(Ω) with s∈(0,1/2]. 

Optimal error estimate of discontinuous Galerkin methods for advection-diffusion-reaction problems with low regularity

In this article, we study the stability of a porous thermoelastic system with Gurtin-Pipkin flux. Under suitable assumptions for the derivative of the heat flux relaxation kernel, we establish the existence and uniqueness of solution by applying the semigroup theory, and prove the exponential stability of system without considering the wave velocity by the means of estimates of the resolvent

/pdf/exponential-stability-for-porous-thermoelastic-systems-with-4c1bc9og.pdf

Exponential stability for porous thermoelastic systems with Gurtin-Pipkin flux

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