Optimal Order Of Convergence Research Articles

The energy-diminishing weak Galerkin finite element method for the computation of ground state and excited states in Bose-Einstein condensates

In this paper, we employ the weak Galerkin (WG) finite element method and the imaginary time method to compute both the ground state and the excited states in Bose-Einstein condensate (BEC) which is governed by the Gross-Pitaevskii equation (GPE). First, we use the imaginary time method for GPE to get the nonlinear parabolic partial differential equation. Subsequently, we apply the WG method to spatially discretize the parabolic equation. This yields a semi-discrete scheme, in which an energy function is explicitly defined. For the case β⩾0, we demonstrate that the energy is diminishing with respect to time t at each time step. Applying the backward Euler scheme for temporal discretization yields a fully discrete scheme. For the case β=0, we provide a mathematical justification, establishing the convergence analysis for the numerical solution of the ground state. Moreover, based on the theory of solving eigenvalue problems using the WG method, we present the error estimates between the ground state and its numerical solution under the H1 and L2 norms. Numerical experiments are provided to illustrate the effectiveness of the proposed schemes. Moreover, the results indicate that our method also can compute the first excited state, achieving optimal convergence orders.

Two Numerical Schemes Via an Auxiliary Function for Nonlocal Problems With Integrable Kernels

ABSTRACTIn this study, we explore volume constrained nonlocal problem with integrable kernels. A central challenge addressed is the discontinuity of true solutions across the boundary, posing difficulties for designing efficient numerical methods. To address this issue, we introduce an auxiliary function which serves as the solution of an auxiliary problem. This approach effectively transforms the original problem to a new nonlocal problem with a true solution which is continuous across the boundary. We propose two numerical schemes to solve the new nonlocal problem: one is exclusively applicable to one‐dimensional case, while the other is suitable for general dimensional cases. Experimental results show that both schemes exhibit optimal convergence order under mild condition.

Divergence-free cut finite element methods for Stokes flow

We develop two unfitted finite element methods for the Stokes equations based on Hdiv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^{{{\\,\ extrm{div}\\,}}}$$\\end{document}-conforming finite elements. Both cut finite element methods exhibit optimal convergence order for the velocity, pointwise divergence-free velocity fields, and well-posed linear systems, independently of the position of the boundary relative to the computational mesh. The first method is a cut finite element discretization of the Stokes equations based on the Brezzi–Douglas–Marini (BDM) elements and involves interior penalty terms to enforce tangential continuity of the velocity at interior edges in the mesh. The second method is a cut finite element discretization of a three-field formulation of the Stokes problem involving the vorticity, velocity, and pressure and uses the Raviart–Thomas (RT) space for the velocity. We present mixed ghost penalty stabilization terms for both methods so that the resulting discrete problems are stable and the divergence-free property of the Hdiv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^{{{\\,\ extrm{div}\\,}}}$$\\end{document}-conforming elements is preserved also on unfitted meshes. In both methods boundary conditions are imposed weakly. We show that imposing Dirichlet boundary conditions weakly introduces additional challenges; (1) The divergence-free property of the RT and the BDM finite elements may be lost depending on how the normal component of the velocity field at the boundary is imposed. (2) Pressure robustness is affected by how well the boundary condition is satisfied and may not hold even if the incompressibility condition holds pointwise. We study two approaches of weakly imposing the normal component of the velocity at the boundary; we either use a penalty parameter and Nitsche’s method or a Lagrange multiplier method. We show that appropriate conditions on the velocity space has to be imposed when Nitsche’s method or penalty is used. Pressure robustness can hold with both approaches by reducing the error at the boundary but the price we pay is seen in the condition numbers of the resulting linear systems, independent of if the mesh is fitted or unfitted to the boundary.

Convergence of a stabilized parametric finite element method of the Barrett–Garcke–Nürnberg type for curve shortening flow

The parametric finite element methods of the Barrett–Garcke–Nürnberg (BGN) type have been successful in preventing mesh distortion/ degeneration in approximating the evolution of surfaces under various geometric flows, including mean curvature flow, Willmore flow, Helfrich flow, surface diffusion, and so on. However, the rigorous justification of convergence of the BGN-type methods and the characeterization of the particle trajectories produced by these methods still remain open since this class of methods was proposed in 2007. The main difficulty lies in the stability of the artificial tangential velocity implicitly determined by the BGN methods. In this paper, we give the first proof of convergence of a stabilized BGN method for curve shortening flow, with optimal-order convergence in L 2 L^2 norm for finite elements of degree k ≥ 2 k \geq 2 under the stepsize condition τ ≤ c h k + 1 \tau \leq c h^{k+1} (for any fixed constant c c ). Moreover, we give the first rigorous characterization of the particle trajectories produced by the BGN-type methods for one-dimensional curves, i.e., we prove that the particle trajectories produced by the stabilized BGN methods converge to the particle trajectories determined by a system of geometric partial differential equations which differs from the standard curve shortening flow by a tangential motion. The characterization of the particle trajectories also rigorously explains, for one-dimensional curves, why the BGN-type methods could maintain the quality of the underlying evolving mesh.

A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order α∈(0,1], where the data are given at two interior points, namely x=x1 and x=x2, and the solution is determined for x∈(0,L),0<x1<x2≤L. The problem is challenging since it is severely ill-posed for x∉[x1,x2]. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.

Optimal convergence order for multi-scale stochastic Burgers equation

Optimal convergence order for multi-scale stochastic Burgers equation

A stabilizer free weak Galerkin method with implicit [formula omitted]-schemes for fourth order parabolic problems

In this study, we solve the fourth-order parabolic problem by combining the implicit θ-schemes in time for θ∈[12,1] with the stabilizer free weak Galerkin (SFWG) method. The semi-discrete and full-discrete numerical schemes are proposed. And specifically, the full-discrete scheme is a first-order backward Euler scheme when θ=1, and a second-order Crank–Nicolson scheme for θ=12. Then, we determine the optimal convergence orders of the error in the H2 and L2 norms after analyzing the well-posedness of the schemes. The theoretical findings are validated by numerical experiments.

Uniform convergence of finite element method on Bakhvalov-type mesh for a 2-D singularly perturbed convection–diffusion problem with exponential layers

Analyzing uniform convergence of finite element method for a 2-D singularly perturbed convection–diffusion problem with exponential layers on Bakhvalov-type mesh remains a complex, unsolved problem. Previous attempts to address this issue have encountered significant obstacles, largely due to the constraints imposed by a specific mesh. These difficulties stem from three primary factors: the width of the mesh subdomain adjacent to the transition point, constraints imposed by the Dirichlet boundary condition, and the structural characteristics of exponential layers. In response to these challenges, this paper introduces a novel analysis technique that leverages the properties of interpolation and the relationship between the smooth function and the layer function on the boundary. By combining this technique with a simplified interpolation, we establish the uniform convergence of optimal order k under an energy norm for finite element method of any order k. Numerical experiments validate our theoretical findings.

Nesterov acceleration‐based iterative method for backward problem of distributed‐order time‐fractional diffusion equation

This paper is concerned with the inverse problem of determining the initial value of the distributed‐order time‐fractional diffusion equation from the final time observation data, which arises in some ultra‐slow diffusion phenomena in applied areas. Since the problem is ill‐posed, we propose an iterated regularization method based on the Nesterov acceleration strategy to deal with it. Convergence rates for the regularized approximation solution are given under both the a priori and a posteriori regularization parameter choice rules. It is shown that the proposed method can always yield the order optimal convergence rates as long as the iteration parameter which appears in the Nesterov acceleration strategy is chosen large enough. In numerical aspect, the main advantage of the proposed method lies in its simplicity. Specifically, due to the Nesterov acceleration strategy, only a few number of iteration steps are required to obtain the approximation solution, and at each iteration step, we only need to numerically solve the standard initial‐boundary value problem for the distributed‐order time‐fractional diffusion equation. Some numerical examples including one‐dimensional and two‐dimensional cases are presented to illustrate the validity and effectiveness of the proposed method.

Virtual element method for solving a viscoelastic contact problem with long memory

In this paper, we use the virtual element method to solve a history-dependent hemivariational inequality arising in contact problems. The contact problem concerns the deformation of a viscoelastic body with long memory, subjected to a contact condition with non-monotone normal compliance and unilateral constraints. A fully discrete scheme based on the trapezoidal rule for the discretization of the time integration and the virtual element method for the spatial discretization are analyzed. We provide a unified priori error analysis for both internal and external approximations. For the linear virtual element method, we obtain the optimal order error estimate. Finally, three numerical examples are reported, providing numerical evidence of the theoretically predicted optimal convergence orders.

A rotational velocity-correction projection method for the Micropolar Navier-Stokes equations

In this paper, we introduce a velocity correction projection method for the Micropolar Navier-Stokes Equations. The velocity correction method are adopted to approximate the time derivative term, stability analysis and error estimation of the first-order semi-discrete scheme are proved. At the same time, the optimal error estimate using the technique of dual norm are obtained. In this way, the divergence free of the velocity u can be conserved. Finally, the numerical results show the method has an optimal convergence order. The numerical results are consistent with our theoretical analysis, and our method is effective.

Very High-Order Accurate Discontinuous Galerkin Method for Curved Boundaries with Polygonal Meshes

Preserving the optimal convergence order of discontinuous Galerkin (DG) discretisations in curved domains is a critical and well-known issue. The proposed approach relies on the reconstruction for off-site data (ROD) method developed originally within the finite volume framework. The main advantages are simplicity, since the PDE solver only considers polygonal domains, and versatility, since any type of boundary condition can be imposed. The developed DG–ROD method consists in splitting the boundary conditions treatment and the leading discrete equations from a classical DG formulation into two independent solvers coupled in a simple and efficient iterative procedure. A numerical benchmark is provided to assess the capability of the method with Dirichlet and Neumann boundary conditions prescribed on curved boundaries, demonstrating that the optimal convergence order is effectively achieved.

Determination of an unknown coefficient in the Korteweg–de Vries equation

Abstract In this paper, a space-time spectral method for solving an inverse problem in the Korteweg–de Vries equation is considered. Optimal order of convergence of the semi-discrete method is obtained in L 2 {L^{2}} -norm. The discrete schemes of the method are based on the modified Fourier pseudospectral method in spatial direction and the Legendre-tau method in temporal direction. The nonlinear term is computed via the fast Fourier transform and fast Legendre transform. The method is implemented by the explicit-implicit iterative method. Numerical results are given to show the accuracy and capability of this space-time spectral method.

ENO and WENO cubic quasi-interpolating splines in Bernstein–Bézier form

In this paper we propose the use of C1-continuous cubic quasi-interpolation schemes expressed in Bernstein–Bézier form to approximate functions with jumps. The construction of these schemes is explicit and consists of directly attaching the Bernstein–Bézier coefficients to appropriate combinations of the given data values. This construction can lead to quasi-interpolation schemes with free parameters. This allows to write these schemes of optimal convergence order as a non-negative convex combination of certain quasi-interpolation schemes of lower convergence order. The idea behind that, is to divide the data set used to define a quasi-interpolant of optimal order into subsets, and then define the associated quasi-interpolants. The free parameters facilitate the choice of the convex combination weights. We then apply the WENO approach to the weights to eliminate the Gibbs phenomenon that occurs when we approximate in a non-smooth region. The proposed schemes are of optimal order in the smooth regions and near optimal order is achieved in the neighboring region of discontinuity.

A nonlinear finite volume element method preserving the discrete maximum principle for heterogeneous anisotropic diffusion equations

In this paper, we propose a new nonlinear conservative and maximum-principle-preserving finite volume element method for heterogeneous anisotropic diffusion equations on triangular or strictly convex quadrilateral meshes. First, we decompose the numerical flux of the standard finite volume element method on each dual edge into two parts: main part having a two-point-flux structure, and residual part owning a multi-point-flux structure. Then, according to the sign of the residual part and the distribution of the local extremum, the residual part is modified to a two-point-flux structure with a nonlinear term. The resulting nonlinear scheme not only inherits conservation and accuracy of original scheme, but also possesses a maximum-preserving structure without extra restrictions on the meshes and diffusion coefficients. Several numerical experiments verify that our nonlinear scheme has an optimal convergence order and satisfies the discrete maximum principle.

Nonconforming virtual element methods for velocity-pressure Stokes eigenvalue problem

In this work, we investigate a divergence-free, nonconforming virtual element method for the numerical approximation of the Stokes eigenvalue problems using polygonal unstructured meshes on polygonal domain. By employing an elliptic projection operator, we “enhance” the definition of the virtual element space to make the L2-projection operator computable with optimal order. Furthermore, we prove the optimal order of convergence of the eigenfunction approximation and the double order of convergence of the eigenvalue approximation through compactness arguments for the solution operator (Babuška-Osborn Theory). Finally, we present a set of numerical experiments to assess the good approximation behavior of the method and show its computational performance.

A low-cost, penalty parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.

A modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes

We introduce a new numerical method for solving time-harmonic Maxwell’s equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite elements are replaced by the average of the two discontinuous polynomial functions on the two sides of the polygon, in the modified weak Galerkin (MWG) finite element method. With the dependent inter-element functions, the weak curl and the weak gradient are defined directly on totally discontinuous polynomials. Optimal-order convergence of the method is proved. Numerical examples confirm the theory and show effectiveness of the modified weak Galerkin method over the existing methods.

HDG Method for Nonlinear Parabolic Integro-Differential Equations

Abstract The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ ( h k + 1 ) O(h^{k+1}) when polynomials of degree k ≥ 0 k\geq 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

Numerical analysis of a singularly perturbed 4th order problem with a shift term

We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with a stability result we have estimates for the subsequent numerical analysis. With classical layer adapted meshes we present a numerical method that achieves supercloseness and optimal convergence orders in the associated energy norm. We also consider coarser meshes in view of the weak layers. Some numerical examples conclude the paper and support the theory.