Optimal Order Of Convergence Research Articles

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.

Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media

Abstract The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.

A locally calculable P3‐pressure in a decoupled method for incompressible Stokes equations

AbstractThis article will suggest a new finite element method to find a ‐velocity and a ‐pressure solving incompressible Stokes equations at low cost. The method solves first the decoupled equation for a ‐velocity. Then, using the calculated velocity, a locally calculable ‐pressure will be defined component‐wisely. The resulting ‐pressure is analyzed to have the optimal order of convergence. Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the ‐velocity. Besides, the method overcomes the problem of singular vertices or corners.

A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖uN−INu‖WG is presented, where INu denotes the vertex-edge-cell interpolation of the solution u, and ‖⋅‖WG denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method.

Sharp analysis of [formula omitted] method on graded mesh for time fractional parabolic differential equation

In this paper, we study the L1−2 method for solving numerically a class of time fractional parabolic differential equations on a graded mesh. Based on the nonsmooth regularity assumptions, the stability and convergence of the proposed numerical method are investigated. Using rigorous and sharp theoretical analysis, we obtain the optimal graded parameter r=(3−α)/(1+α) of the graded mesh. Numerical experiments are given to demonstrate that even if the α is small, the graded mesh based on the optimal graded parameter is still able to recover the optimal convergence order of the L1−2 method.

Higher-order galerkin finite element method for nonlinear coupled reaction-diffusion models

In this article, generalized nonlinear coupled reaction-diffusion (NCRD) models are analyzed using the higher-order Galerkin finite element method (FEM). The authors present finite element analysis for one- and two-dimensional NCRD models by applying quadratic Lagrange basis functions. Optimal order convergence in space is proved by applying quadratic FEM in spatial discretization and utilizing the Crank-Nicolson (CN) scheme for time discretization. To address the nonlinearity of NCRD models, CN extrapolation and predictorcorrector schemes are employed. The stability analysis for generalized NCRD models and the applied CN scheme are discussed by utilizing the eigenvalues method. Finally, to assess the accuracy and efficiency of the proposed scheme and to capture different patterns of NCRD models, various one- and two-dimensional models are considered. The obtained results are numerically and graphically compared with existing literature. It is observed that quadratic basis functions in FEM improve accuracy and efficiency of the results.

Numerical discretization for Fisher-Kolmogorov problem with nonlocal diffusion based on mixed Galerkin BDF2 scheme

Nonlocal problems involving fourth-order terms pose several difficulties such as numerical discretization and its related convergences analysis. In this paper, the well-posedness of the extended Fisher-Kolmogorov equation with nonlocal diffusion is first analyzed using the Faedo-Galerkin technique and the classical compactness arguments. Moreover, we adopt a BDF2 scheme for time discretization and a mixed Galerkin scheme for spatial discretization. Then, we derive the optimal order convergence rates of the fully discrete system. Finally, some numerical simulations and convergence results are provided to confirm the theoretical results and the accuracy of the proposed scheme.

A robust weak Galerkin finite element method for two parameter singularly perturbed parabolic problems on nonuniform meshes

In this study, we proposed a weak Galerkin finite element method (WG-FEM) for solving two-parameter singularly perturbed parabolic problems (TP-SPPPs) of convection–diffusion–reaction type on nonuniform mesh. The WG-FEM approach incorporates an interpolation operator IN and achieves optimal order convergence in an energy-like norm for continuous and fully discrete schemes. In the fully-discrete analysis, we combined WG-FEM for spatial space on a Shishkin-type mesh with the Crank–Nicolson scheme for temporal discretization on an equidistant mesh. Since the energy norm is insufficient to capture the behavior of the boundary layer functions accurately, we have derived the optimal order of convergence in a strong balanced norm. This norm exhibits an order of O(N−1lnN)k in spatial convergence and second-order convergence in time when using an L2-projection QN as an intermediate operator. We conducted numerical examples to validate the proposed method’s theoretical findings. The results of these experiments confirmed the theoretical conclusions and demonstrated the robustness of the proposed method.

Error Analysis of the Nonuniform Alikhanov Scheme for the Fourth-Order Fractional Diffusion-Wave Equation

This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a weak singularity at the initial time. Using the order reduction method, the fourth-order fractional diffusion-wave equation can be rewritten as a coupled system of low order, which is approximated by the nonuniform Alikhanov scheme in time and the finite difference method in space. Furthermore, the H2-norm stability result is obtained. With the help of this result and a priori bounds of the solution, an α-robust error estimate with optimal convergence order is derived. In order to further verify the accuracy of our theoretical analysis, some numerical results are provided.

An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain

Abstract The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche’s method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity–pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $L^{2}(H^{1})$-type norm for the pressure component.

A stable and high-order numerical scheme with discrete DtN-type artificial boundary conditions for a 2D peridynamic diffusion model

In this work, a stable and high-order numerical scheme with discrete DtN-type artificial boundary conditions (ABCs) is designed for solving a peridynamic diffusion model on the whole two dimensional domain. To do so, we first use a high-order quadrature-based finite difference scheme to approximate the spatially nonlocal operator and use the BDF2 scheme to approximate the temporal direction. For the resulting fully discrete system, we apply the nonlocal potential theory to obtain the discrete Dirichlet-to-Dirichlet (DtD)-type ABCs. After that, we further derive the Dirichlet-to-Neumann (DtN)-type ABCs based on the nonlocal Neumann data defined from the discrete nonlocal Green's first identity. For the stability and convergence analysis, we reformulate the BDF2 operator into a discrete convolution sum form, then use the technique of the discrete orthogonal convolution kernels to obtain the optimal convergence order. Finally, numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

A two grid partition of unity finite element method for electrically conducting incompressible fluid flows

In this article, a two grid partition of unity finite element method is proposed and investigated for electrically conducting incompressible fluid flows. This algorithm involves solving a much smaller nonlinear problem on a coarse grid, utilizing a partition of unity to decompose reasonably the residual problem into a series of independent subproblems on a fine grid, and carrying out a further coarse correction on the coarse grid. Rigorously theoretical analysis is presented and convergence results indicate that the method could reach the optimal convergence orders with proper configurations between the coarse mesh size H and the fine mesh size h. Finally, some numerical results are reported to verify our theoretical findings.

Full weak Galerkin finite element discretizations for poroelasticity problems in the primal formulation

This paper develops novel finite element solvers for linear poroelasticity problems on quadrilateral meshes. These solvers are based on the primal formulations of linear elasticity and Darcy flow. Specifically, the fluid pressure and solid displacement are approximated by scalar- or vector-valued polynomials of degree k≥0 separately in element interiors and on edges. The discrete weak gradients of these shape functions are established in the broken (vector- or matrix-version) Arbogast–Correa spaces for approximations of the classical gradients in the variational forms. These weak Galerkin spatial discretizations are combined with the implicit Euler or Crank–Nicolson temporal discretizations to develop locking-free numerical solvers that have optimal order (k+1) convergence rates in pressure, velocity, displacement, stress, and dilation. Rigorous analysis is presented and illustrated by numerical experiments on popular test cases.

A mimetic interpolation-free cell-centered finite volume scheme for the 2D and 3D heterogeneous anisotropic diffusion equations

We propose a mimetic interpolation-free cell-centered finite volume scheme for diffusion equations on arbitrary two-dimensional (2D) and three dimensional (3D) unstructured, possibly non star-shaped or nonplanar meshes. This scheme borrows ideas from the well known mimetic finite difference methods. This interpolation-free means that we do not need to introduce auxiliary unknowns in the process of constructing the scheme, and the final scheme has only cell-centered unknowns. After carefully handling continuity of solutions and normal fluxes, we construct piecewise linear approximations on each cell. This makes our scheme applicable to arbitrary discontinuities and arbitrary mesh topologies. Interpolation-free makes the program easy to implement and the 2D and 3D programs have a unified framework. It is worth mentioning that the traditional star-shaped mesh assumption has been abolished, and the new scheme has more relaxed restrictions on meshes. Some representative and even challenging numerical examples verify that the scheme is robust and almost has the optimal convergence order on some benchmark polygonal and polyhedral meshes.

Structure preservation in high-order hybrid discretisations of potential-driven advection-diffusion: linear and nonlinear approaches

<abstract><p>We are interested in the high-order approximation of anisotropic, potential-driven advection-diffusion models on general polytopal partitions. We study two hybrid schemes, both built upon the Hybrid High-Order technology. The first one hinges on exponential fitting and is linear, whereas the second is nonlinear. The existence of solutions is established for both schemes. Both schemes are also shown to possess a discrete entropy structure, ensuring that the long-time behaviour of discrete solutions mimics the PDE one. For the nonlinear scheme, the positivity of discrete solutions is a built-in feature. On the contrary, we display numerical evidence indicating that the linear scheme violates positivity, whatever the order. Finally, we verify numerically that the nonlinear scheme has optimal order of convergence, expected long-time behaviour, and that raising the polynomial degree results, also in the nonlinear case, in an efficiency gain.</p></abstract>

Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O(λN−γ) in space and O(τγ) in time, where γ∈(0,1] from the assumption ∥Aγ−12Q12∥L2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings.

Variational multiscale stabilized finite element analysis of transient MHD Stokes equations with application to multiply driven cavity flow

The transient Stokes magnetohydrodynamic (Stokes MHD) equations are thoroughly investigated in this paper using the variational subgrid multiscale stabilized finite element approach. In order to get the optimal order of convergence, appropriate stabilization parameter expressions have been derived using Fourier analysis. The theoretical convergence analysis of the subgrid algebraic stabilized numerical scheme is also presented in this study. The numerical studies on multiply-driven cavity flow establish the credibility of the variational algebraic subgrid multiscale stabilized finite element approach. Further, the theoretically determined convergence rate is numerically validated. For different Hartmann, Reynolds, and magnetic force inclination angle values, the flow fields are traced.

An Eulerian finite element method for tangential Navier-Stokes equations on evolving surfaces

The paper introduces a geometrically unfitted finite element method for the numerical solution of the tangential Navier–Stokes equations posed on a passively evolving smooth closed surface embedded in R 3 \mathbb {R}^3 . The discrete formulation employs finite difference and finite elements methods to handle evolution in time and variation in space, respectively. A complete numerical analysis of the method is presented, including stability, optimal order convergence, and quantification of the geometric errors. Results of numerical experiments are also provided.

BDM H(div) weak Galerkin finite element methods for Stokes equations

We add new inter-element variables to the BDM H(div) finite elements on triangular and tetrahedral meshes for the Stokes equations. With the weak gradient, we enforce weakly the tangential continuity of H(div) finite element velocity fields while not relaxing the normal continuity. The new method produces divergence-free H(div) finite element solutions of optimal-order convergence. Numerical examples are provided, confirming the theory and in particular, the pressure-robustness of the method.