Optimal Approximation Order Research Articles

Analytical and numerical studies of a cancer invasion model with nonlocal diffusion

Abstract This article investigates the implicit Euler–Galerkin finite element method applied to a cancer invasion model with nonlocal diffusion. This model describes the normal cell density, tumor cell density, excess H + ion concentration, extracellular matrix, and extracellular matrix active metalloproteinases. The primary goal of this work is to utilize numerical solutions to comprehend cancer invasion and transmission within the patient. The Faedo–Galerkin approximation method is initially used to establish the existence of a weak solution, followed by deriving the a priori error estimates of optimal order. Additionally, the implicit Euler–Galerkin finite element method is employed as a numerical tool for solving the given cancer invasion parabolic system. Furthermore, a priori error bounds and convergence estimates for the fully discrete problem are derived. Finally, numerical simulations are conducted to validate the theoretical conclusions and enhance the understanding of how cancer spreads within the patient.

Approximation error estimates by noise‐injected neural networks

One‐hidden‐layer feedforward neural networks are described as functions having many real‐valued parameters. Approximation properties of neural networks are established (universal approximation property), and the approximation error is related to the number of parameters in the network. The essentially optimal order of approximation error bounds was already derived in 1996. We focused on the numerical experiment that indicates the neural networks whose parameters contain stochastic perturbations gain better performance than ordinary neural networks and explored the approximation property of neural networks with stochastic perturbations. In this paper, we derived the quantitative order of variance of stochastic perturbations to achieve the essentially optimal approximation order and verified the justifiability of our theory by numerical experiments.

On the Optimal Order Approximation of the Partition of Unity Finite Element Method

On the Optimal Order Approximation of the Partition of Unity Finite Element Method

A flux‐based HDG method

AbstractIn this article, we present a flux‐based formulation of the hybridizable discontinuous Galerkin (HDG) method for steady‐state diffusion problems and propose a new method derived by letting a stabilization parameter tend to infinity. Assuming an inf‐sup condition, we prove its well‐posedness and error estimates of optimal order. We show that the inf‐sup condition is satisfied by some triangular elements. Numerical results are also provided to support our theoretical results.

Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method’s key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete H1 and L2 norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

On discrete ground states of rotating Bose–Einstein condensates

The ground states of Bose–Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross–Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the L 2 L^2 - and H 1 H^1 -norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler–Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.

MIXED VOLUME ELEMENT-CHARACTERISTIC MIXED VOLUME ELEMENT OF NUMERICAL SIMULATION OF NUCLEAR CONTAMINATION TREATMENT AND ITS CONVERGENCE ANALYSIS

We consider a nonlinear system with boundary-initial value conditions of convection-diffusion partial differential equations describing nuclear waste disposal contamination in porous media. The flow pressure is determined by an elliptic equation, the concentrations of brine and radionuclide are formulated by convection-diffusion equations, and the transport of temperature is defined by a heat equation. The transport pressure appears in the concentration equations and heat equation accompanying with Darcy velocity, and controls their processes. The flow equation is solved by the conservative method of mixed volume element and the accuracy of Darcy velocity is improved one order. The method of characteristic mixed volume element is applied to solve the concentrations and the heat, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm high computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while it has smaller time-truncation error and higher order of accuracy. The mixed volume element method has law of conservation on every element to treat the diffusion and it can obtain numerical solution is of the concentration and adjoint vectors. Using the theory and technique of priori estimate of differential equations, we derive an optimal second order estimate in l2 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve such an international famous problem.

A generalized weak Galerkin method for Oseen equation

In this work, the authors introduce a generalized weak Galerkin (gWG) finite element method for the evolutionary Oseen equation. The gWG method is based on a new framework for approximating the gradient operator. Both a semi-discrete and a fully-discrete numerical schemes are developed and analyzed for their convergence, stability, and error estimates. The backward Euler discretization is employed in the design of the fully-discrete scheme. Error estimates of optimal order are established mathematically, and they are validated numerically with some benchmark examples.

Continuous interior penalty stabilization for divergence-free finite element methods

Abstract In this paper, we propose, analyze and test numerically a pressure-robust stabilized finite element for a linearized problem in incompressible fluid mechanics, namely, the steady Oseen equation with low viscosity. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac 12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided.

Weak Galerkin finite element method for linear poroelasticity problems

In this paper, we develop a weak Galerkin (WG) finite element method for a linear poroelasticity model where weak divergence and weak gradient operators defined over discontinuous functions are introduced. We establish both the continuous and discrete time WG schemes, and obtain their optimal convergence order estimates in a discrete H1 norm for the displacement and in H1 and L2 norms for the pressure. Finally, we present some numerical experiments on different kinds of meshes to illustrate the theoretical error estimates, and furthermore verify the locking-free property of our proposed method.

Hyperspectral inversion of soil water and salt information based on fractional order derivative technology.

Accurate and efficient acquisition of soil water and salt information is a prerequisite for the improvement and sustainable utilization of saline lands. With the ground field hyperspectral reflectance and the measured soil water-salt content as data sources, we used the fractional order differentiation (FOD) technique to process hyperspectral data (with a step length of 0.25). The optimal FOD order was explored at the correlation level of spectral data and soil water-salt information. We constructed two-dimensional spectral index, support vector machine regression (SVR) and geographically weighted regression (GWR). The inverse model of soil water-salt content was finally evaluated. The results showed that FOD technique could reduce the hyperspectral noise and explore the potential spectral information to a certain extent, improve the correlation between spectrum and characteristics, with the highest correlation coefficients of 0.98, 1.35 and 0.33. The combination of characteristic bands screened by FOD and two-dimensional spectral index were more sensitive to characteristics than one-dimensional bands, with the optimal responses of order 1.5, 1.0 and 0.75. The optimal combinations of bands for maximum absolute correction coefficient of SMC were 570, 1000, 1010, 1020, 1330 and 2140 nm, pH were 550, 1000, 1380 and 2180 nm, and salt content were 600, 990, 1600 and 1710 nm, respectively. Compared with the original spectral reflectance, the validation coefficients of determination (Rp2) of the optimal order estimation models for SMC, pH, and salinity were improved by 1.87, 0.94 and 0.56, respectively. The overall GWR accuracy in the proposed model was better than SVR, where the GWR optimal order estimation models Rp2 were 0.866, 0.904 and 0.647, and the relative per-centage difference were 3.54, 4.25 and 1.86, respectively. The overall spatial distribution of soil water and salt content in the study area was characterized by low in the west and high in the east, with more serious soil alkalinization problems in the northwest and less severe in the northeast. The results would provide scientific basis for the hyperspectral inversion of soil water and salt in the Yellow River Irrigation Area and a new strategy for the implementation and management of precision agriculture in saline soil areas.

Quasi-interpolation in Spline Spaces: Local Stability and Approximation Properties

In this work we analyze the approximation error in Sobolev norms for quasi-interpolation operators in spline spaces. We establish in a general way the hypotheses on a quasi-interpolant to achieve the optimal order of approximation. Finally, we propose simple but general constructions of such operators that satisfy the established hypotheses and illustrate their performance through some numerical tests.

$ W^{1, \infty} $-seminorm superconvergence of the block finite element method for the five-dimensional Poisson equation

<abstract><p>This study focused on the superconvergence of the finite element method for the five-dimensional Poisson equation in the $ W^{1, \infty} $-seminorm. Specifically, we investigated the block finite element method, which is a tensor-product finite element approach applied to regular rectangular partitions of the domain. First, we introduced the finite element scheme for the equation and discussed various functions related to it, along with their properties. Next, we proposed a weight function and established its important properties, which play a crucial role in the theoretical analysis. By utilizing the properties of the weight function and employing weighted-norm analysis techniques, we derived an optimal order estimate in the $ W^{2, 1} $-seminorm for the discrete derivative Green's function (DDGF). Furthermore, we provided an interpolation fundamental estimate of the second type, also known as the weak estimate of the second type, for the block finite element. This weak estimate is based on a five-dimensional interpolation operator of the projection type. Finally, by combining the derived $ W^{2, 1} $-seminorm estimate for the DDGF and the weak estimate for the block finite element, we obtained a superconvergence estimate for the block finite element approximation in the pointwise sense of the $ W^{1, \infty} $-seminorm. The correctness of the theoretical results was demonstrated through a numerical example.</p></abstract>

Quasi-factorization and multiplicative comparison of subalgebra-relative entropy

Purely multiplicative comparisons of quantum relative entropy are desirable but challenging to prove. We show such comparisons for relative entropies between comparable densities, including the relative entropy of a density with respect to its subalgebraic restriction. These inequalities are asymptotically tight in approaching known, tight inequalities as perturbation size approaches zero. Based on these results, we obtain a kind of inequality known as quasi-factorization or approximate tensorization of relative entropy. Quasi-factorization lower bounds the sum of a density’s relative entropies to several subalgebraic restrictions in terms of its relative entropy to their intersection’s subalgebraic restriction. As applications, quasi-factorization implies uncertainty-like relations, and with an iteration trick, it yields decay estimates of optimal asymptotic order on mixing processes described by finite, connected, undirected graphs.

Quasi-interpolation for analysis-suitable T-splines

We propose a novel local approximation method for analysis-suitable T-spline (AS T-spline) spaces via quasi-interpolation. The quasi-interpolants are defined as linear combination of the approximated function's values at appropriately chosen points. Benefited from the inherent nice properties of AS T-splines, the proposed quasi-interpolants can reproduce polynomials up to the same degree of AS T-spline spaces and can provide optimal approximation order. Some numerical examples of specific quasi-interpolants for bi-cubic AS T-splines are investigated to show the stability and efficiency.

Optimal [formula omitted](div) flux approximations from the Primal Hybrid Finite Element Method on quadrilateral meshes

In this work, we propose and analyze an element-wise H(div,Ω)-conforming computational strategy to post-process the flux vector from the approximated solution of a second-order elliptic equation, obtained by the Primal Hybrid Finite Element Method on regular meshes of convex quadrilaterals generated by bilinear isomorphisms. The recovery strategy makes use of the Arnold–Boffi–Falk (ABFt, t≥0) spaces, leading to locally conservative approximated fluxes with continuous normal components. The proposed method is proven to provide optimal order approximations in H(div,Ω), in the sense that both the flux and its divergence achieve optimal order ht+1 in the L2(Ω)-norm, on quadrilateral meshes. We also show that, within the recovery strategy, it is possible to achieve higher-order approximations for the divergence of the flux by increasing the index t of the local spaces, with almost no extra overall computational cost. Numerical results on affine and bilinear quadrilateral meshes are presented, confirming the theoretical predictions.

A Modified Fifth-Order WENO-HLLC Riemann Solver for Transonic Viscous Flows around Helicopter Rotor in Hover

This paper presents a modified fifth-order WENO-HLLC Riemann solver for the transonic compressible viscous flows around the helicopter rotor in hover. The HLLC approximate Riemann solver is proposed to discrete the convection term involving grid velocity of the Navier-Stokes equations. In order to solve the interface flow accurately, a modified fifth-order WENO scheme is presented by designing the smoothness indicators based on L 1 norm measurement. The improved WENO scheme can provide the optimal approximation order even at critical points. Numerical accuracy and robustness are validated by several benchmark inviscid flow problems. Then the numerical properties of the WENO-HLLC solver in conjunction with the implicit LU-SGS time integration method with high efficiency are further validated by simulating transonic viscous flows over RAE2822 airfoil and ONERA-M6 wing. The results show that the accuracy of calculating shock, discontinuity, and the vortex is significantly improved. Finally, the method is developed to compute the transonic vortex flow around the helicopter rotor with a domain discretized by overset grids. The results indicate that the proposed method is very robust and effective in acquiring high resolution for vortex wake.

Optimal convergence of arbitrary Lagrangian–Eulerian iso-parametric finite element methods for parabolic equations in an evolving domain

Abstract An optimal-order error estimate is presented for the arbitrary Lagrangian–Eulerian (ALE) finite element method for a parabolic equation in an evolving domain, using high-order iso-parametric finite elements with flat simplices in the interior of the domain. The mesh velocity can be a linear approximation of a given bulk velocity field or a numerical solution of the Laplace equation with specified boundary value matching the velocity of the boundary. The optimal order of convergence is obtained by comparing the numerical solution with the ALE-Ritz projection of the exact solution, and by establishing an optimal-order estimate for the material derivative of the ALE-Ritz projection error.

Optimal $W^{1,\infty}$-estimates for an isoparametric finite element discretization of elliptic boundary value problems

We consider an elliptic boundary value problem on a domain with regularboundary and discretize it with isoparametric finite elements of order$k\geq1$.We show optimal order of convergence of the isoparametric finiteelement solution in the $W^{1,\infty}$-norm. As an intermediate step,we derive stability and convergence estimates of optimal order $k$ fora (generalized) Ritz map.

Construction of a normalized basis of a univariate quadratic $C^1$ spline space and application to the quasi-interpolation

In this paper, we use the finite element method to construct a new normalized basis of a univariate quadratic $C^1$ spline space. We give a new representation of Hermite interpolant of any piecewise polynomial of class at least $C^1$ in terms of its polar form. We use this representation for constructing several superconvergent and super-superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.