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The Scott–Vogelius element is a popular finite element for the discretization of the Stokes equations which enjoys inf-sup stability and gives divergence-free velocity approximations. However, it is well known that the convergence rates for the discrete pressure deteriorate in the presence of certain critical vertices in a triangulation of the domain. Modifications of the Scott–Vogelius element such as the recently introduced pressure-wired Stokes element also suffer from this effect. In this paper we introduce a simple modification strategy for these pressure spaces that preserves the inf-sup stability while the pressure converges at an optimal rate. [ABSTRACT FROM AUTHOR]

In this paper, we study adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two-dimensional numerical examples for illustration of efficiency and accuracy. Theoretical findings consist in the vanishing limit of a subsequence of estimators and the convergence of the relevant subsequence of adaptively-generated solutions to a solution to the continuous optimality system. [ABSTRACT FROM AUTHOR]

The Poisson–Boltzmann equation, which incorporates the source of the Dirac distribution, has been widely applied in predicting the electrostatic potential of biomolecular systems in solution. In this paper we discuss and analyse the virtual element method for the Poisson–Boltzmann equation on general polyhedral meshes. Nearly optimal error estimates, approaching the best possible accuracy, are achieved for the virtual element approximation in both the L 2-norm and H 1-norm, even when the solution of the entire domain has low regularity. The efficiency of the virtual element method and the validity of the proposed theoretical prediction are confirmed through numerical experiments conducted on various polyhedral meshes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel unfitted finite element method to solve the quad-curl interface problem. We adapt Nitsche’s method for curl curl -conforming elements and double the degrees of freedom on interface elements. To ensure stability, we incorporate ghost penalty terms and a discrete divergence-free term. We establish the well-posedness of our method and demonstrate an optimal error bound in the discrete energy norm. We also analyze the stiffness matrix’s condition number. Our numerical tests back up our theory on convergence rates and condition numbers. [ABSTRACT FROM AUTHOR]

This article examines the convergence of the two-step backward difference formula (BDF2)-discontinuous Galerkin (DG) scheme in the context of a nonlocal reaction–diffusion equation. Our methodology integrates BDF2 for temporal discretization with a DG finite element method for spatial discretization, enabling a comprehensive analysis of the nonlocal diffusion equation. We establish optimal-order convergence rates for the fully discrete system. To assess the effectiveness and precision of the proposed scheme, we provide numerical simulations alongside convergence results. [ABSTRACT FROM AUTHOR]

In this paper, an Arrow-Hurwicz iterative finite element method based on charge-conservation for solving the stationary inductionless magnetohydrodynamic equations is proposed and analyzed. The current density and electric potential are discretized by H (div , Ω) × L 2 (Ω) -conforming finite element pairs, respectively, which maintains charge-conservation. The hydrodynamic unknowns are discretized by stable velocity-pressure finite element pairs. Furthermore, we use the Arrow-Hurwicz iterative method to decouple the velocity and pressure, which leads to that there is no large-scale saddle point system that needs to be solved at each iteration step except for the determination of the initial functions. It has been proven that the proposed method converges geometrically with a contraction number independent of the mesh width. Numerical results are presented to verify the results of theoretical analysis and the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the conservation characteristic difference method based on the second-order modified upwind scheme is analyzed for solving the two-dimensional convection-diffusion equations by combining the splitting technique. Along each directional, the intermediate numerical solutions are first computed by the piecewise parabolic method (PPM), where the endpoint values over each domain are solved by the second-order modified upwind scheme. Then, the solutions are computed by the splitting implicit characteristic difference method. By some auxiliary lemmas, we can prove that our scheme is stable in L 2 -norm. The optimal error estimate is given and the scheme is convergent of second order in space. Numerical experiments are given to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

We suggest using a GMRES preconditioner to speed up the convergence of Krylov subspace methods for solving linear equations with a block structure. These types of equations commonly arise in unfitted finite element discretizations of the Darcy flow problem in a fracture network. We have thoroughly examined the spectrum of the preconditioned matrix to demonstrate its effectiveness. Additionally, we have conducted numerical experiments to highlight how well the preconditioners work with flexible-GMRES when solving linear equations from benchmark test problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct optimal error estimates of second-order stabilized scheme for the incompressible magnetohydrodynamic(MHD) system. For this purpose, we first construct first- and second-order semi-discrete schemes in which the time derivative term is treated by the first-order backward Euler method and the second-order backward difference formulation, respectively. Moreover, the nonlinear terms are treated by semi-implicit method, and the coupling of velocity and pressure is decoupled by a Gauge–Uzawa method. Thus, the schemes achieve decoupling of velocity, magnetic field and pressure. Most importantly, the proposed schemes do not need to deal with the artificial boundary conditions on pressure and do not need to give an initial value of pressure. Then, the unconditional stability of the two schemes is demonstrated. Furthermore, through rigorous error analysis, we provide optimal convergence orders for all unknowns. Finally, some numerical experiments demonstrate the accuracy and effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we resort to proper orthogonal decomposition (POD) to reduce the dimension of unknown coefficient vectors of finite element (FE) solutions of two-grid Crank-Nicolson FE (TGCNFE) method for the nonlinear parabolic equation and develop a new two-grid reduced-dimension extrapolated CNFE (TGRDECNFE) method. To this end, we first develop a new TGCNFE method for the nonlinear parabolic equation and prove the existence, unconditional stability, and error estimates for the TGCNFE solutions. We then develop the TGRDECNFE method for the nonlinear parabolic equation by using the POD technique to lower the dimensionality of unknown coefficient vectors in the TGCNFE solutions and adopt the matrix analysis to analyze the existence, unconditional stability, and errors for the TGRDECNFE solutions. We finally employ two numerical examples to confirm our theoretical results and to explain the advantage of TGRDECNFE method. It has been verified on the whole network that both TGCNFE method and TGRDECNFE method herein are completely different from the existing works. Therefore, they are fire-new. [ABSTRACT FROM AUTHOR]

A variational formulation based on velocity and stress is developed for linear fluid–structure interaction problems. The well-posedness and energy stability of this formulation are established. A hybridizable discontinuous Galerkin method is employed to discretize the problem. An hp-convergence analysis is performed for the resulting semi-discrete scheme. The Crank–Nicolson method is used for temporal discretization, and the convergence properties of the fully discrete scheme are examined. Numerical experiments are presented to validate the theoretical results and demonstrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

The manuscript focuses on establishing an optimal accurate parameter-uniform scheme for singularly perturbed reaction-diffusion type problems in two dimensions. Along with the higher dimensions, these model problems have other complexities like the presence of multiple boundary layers because of the perturbation parameter. To handle these complexities, at first, the model problem is split into two lower-dimensional problems using the Peaceman–Rachford splitting algorithm based on the Crank–Nicolson scheme on a uniform mesh. Further, to be consistent with the second-order accuracy in the temporal direction, the spatial direction is addressed using the central difference scheme on some layer rectifying Shishkin-type meshes namely, the standard Shishkin mesh and the Bakhvalov-Shishkin mesh. The study extends with the application of the proposed scheme on a comparatively new Shishkin-type mesh, namely the Modified-Bakhvalov–Shishkin mesh. The use of the Bakhvalov–Shishkin-mesh and the Modified-Bakhvalov–Shishkin-mesh rectifies the logarithmic effect on the accuracy because of the standard Shishkin mesh. The scheme is made more computationally desirable by the use of robust Thomas algorithm. A global second-order accuracy is affirmed. All the theoretical claims are further corroborated in practice with some valid numerical tests and comparison results. In support of the claims, numerical experiments are also performed on model problems with larger time delays. [ABSTRACT FROM AUTHOR]

In this paper, we address a two-parameter singularly perturbed convection-reaction-diffusion 2-D problem. We also consider that the convection and source terms are discontinuous in space. Due to these discontinuities and the presence of perturbation parameters, solutions to such problems show boundary and interior layers. In this study, we have carried out a numerical approach using a finite-difference technique with an appropriate layer-adapted piecewise uniform Shishkin mesh. Some examples are presented which show the best performance of the proposed method and its agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use energy space regularizations in the objective functional, while the equivalent representations of the energy space norms, i.e., the H - 1 (Ω) -norm for the distributed control, the H 1 / 2 (Γ) -norm for the Dirichlet control and the H - 1 / 2 (Γ) -norm for the Neumann control, enable us to transform the optimal control problem into an elliptic variational inequality involving only the state variable. The elliptic variational inequalities are second order for the three cases, and include additional equality constraints for Dirichlet or Neumann boundary control problems. Standard C 0 finite element methods can be used to solve the resulted variational inequalities. We provide preliminary a priori error estimates for the new method for solving distributed control problems. Extensive numerical experiments are carried out to validate the accuracy of the new method. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyze a weighted residual a posteriori error estimate for a sparse optimal control problem. The problem involves a non-differentiable cost functional, a state equation with an integral fractional Laplacian, and control constraints. We employ subdifferentiation in non-differentiable convex analysis to obtain first-order optimality conditions. Piecewise linear polynomials are utilized to approximate the solutions of the state and adjoint equations. The control variable is discretized by the variational discretization method. Upper bounds for the a posteriori error estimate of the finite element approximation of the optimal control problem are derived. One challenge in devising a posteriori error estimators is poor properties of the residual. Namely, it is not necessarily in L 2 (Ω) . To address this issue, the weighted residual estimator incorporates additional weight computed as the power of the distance from the mesh skeleton. Furthermore, we propose an h-adaptive algorithm driven by the a posteriori error estimator, utilizing the Dörfler labeling criterion. The convergence analysis results show that the approximation sequence generated by the adaptive algorithm converges at the optimal algebraic rate. Finally, numerical experiments are conducted to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a new semi-implicit stochastic multiscale method for the radiative heat transfer problem with additive noise fluctuation in composite materials. In the proposed method, the strong nonlinearity term induced by heat radiation is first approximated, by a semi-implicit predictor-corrected numerical scheme, for each fixed time step, resulting in a spatially random multiscale heat transfer equation. Then, the infinite-dimensional stochastic processes are modeled and truncated using a complete orthogonal system, facilitating the reduction of the model's dimensionality in the random space. The resulting low-rank random multiscale heat transfer equation is approximated and computed by using efficient spatial basis functions based multiscale method. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial multiscale properties, the high-dimensional randomness and the strong nonlinearity of the solution, so they can be overcome separately using different strategies. The convergence analysis is carried out, and the optimal rate of convergence is also obtained for the proposed semi-implicit stochastic multiscale method. Numerical experiments on several test problems for composite materials with various microstructures are also presented to gauge the efficiency and accuracy of the proposed semi-implicit stochastic multiscale method. [ABSTRACT FROM AUTHOR]

This paper proposes and investigates the two-grid stabilized lowest equal-order finite element method for the time-independent dual-permeability-Stokes model with the Beavers-Joseph-Saffman-Jones interface conditions. This method is mainly based on the idea of combining the two-grid and the two local Gauss integrals for the dual-permeability-Stokes system. In this technique, we use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the dual-permeability-Stokes model using the lowest equal-order finite element quadruples. In the two-grid scheme, the global problem is solved using the standard P 1 - P 1 - P 1 - P 1 finite element approximations only on a coarse grid with grid size H. Then, a coarse grid solution is applied on a fine grid of size h to decouple the interface terms and the mass exchange terms for solving the three independent subproblems such as the Stokes equations, microfracture equations, and the matrix equations on the fine grid. On the other hand, microfracture and matrix equations are decoupled through the mass exchange terms. The weak formulation is reported, and the optimal error estimate is derived for the two-grid schemes. Furthermore, the numerical results validate that the two-grid stabilized lowest equal-order finite element method is effective and has the same accuracy as the coupling scheme when we choose h = H 2 . [ABSTRACT FROM AUTHOR]

In this work, we extend recent research on the fully mixed virtual element method based on Banach spaces for the stationary Boussinesq equation to suggest and analyze a new mixed-virtual element technique for the stationary generalized Boussinesq equations, where viscosity and thermal conductivity are temperature-dependent. Besides the original thermo-fluid variables, the pseudostress and vorticity tensors of the fluid and the pseudoheat vector of the thermal system are introduced as auxiliary unknowns, and the incompressibility condition is then used to remove the pressure, which is later computed using a postprocessing formula. The resulting highly coupled formulation is then written equivalently as uncoupled problems thanks to a fixed-point strategy, so that the classical Banach Theorem, combined with the corresponding Babuška–Brezzi theory, the Banach–Nečas–Babuška Theorem, smallness-of-data assumptions, and a slight higher-regularity assumption for the exact solution, are employed to establish the unique solvability of the continuous formulation. The virtual element discretization of the uncoupled problems is based on the H (div 6 / 5) - and H (div 6 / 5) -conforming virtual element techniques. The discrete analysis is conducted in a similar manner by establishing the discrete inf-sup condition and using the discrete versions of the aforementioned theorems to demonstrate the existence of the discrete solution and derive its stability estimates. In addition, a priori error estimates are established by utilizing the Céa estimate and a suitable assumption on data for all variables in their natural norms showing an optimal rate of convergence. Finally, several numerical examples are presented to illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a novel first-order, fully decoupled, unconditionally stable time discretization scheme for the MHD system with variable density. This scheme successfully decouples all the coupling terms by combining the gauge-Uzawa method and the scalar auxiliary variable (SAV) method. And we prove its unconditional energy stability. Then we give the first-order finite element scheme and its implementation. Furthermore, we perform a rigorous error analysis of the proposed numerical scheme. Finally, we perform some numerical experiments to demonstrate the effectiveness of the decoupling scheme. [ABSTRACT FROM AUTHOR]

In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term b ∈ [ L ∞ ⁢ (Ω) ] d and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only H 1 -regularity, which extends the result [J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 2008, 5, 2524–2550]. The theoretical results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

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) when polynomials of degree k ≥ 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. [ABSTRACT FROM AUTHOR]

For the tempered nonlocal Laplacian, an efficient and accurate numerical evaluation in multi-dimensions is challenging due to the nature of a singular integral. In this research, we present a new approximation using the generating function or multiplier approximation theory to discretize the multi-dimensional tempered nonlocal Laplacian defined by a hypersingular integral. We show that the approximation can uniformly achieve the convergence of the second order, independent of the operator parameters. Using the proposed approximation, we construct a simple and easy-to-implement finite difference scheme to solve elliptic and parabolic equations with the tempered nonlocal Laplacian. We provide the analysis for convergence and stability of the scheme for the elliptic and parabolic equations. We also present a fast algorithm with quasi-linear complexity of the scheme for computing the tempered operator. Several numerical examples demonstrate the accuracy and efficiency of our algorithm and verify our theory. [ABSTRACT FROM AUTHOR]

In this article, a fourth-order finite-difference ghost-point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high-order extension of the second-order ghost method introduced earlier by the authors. Three different discretizations are considered, which differ in the stencil that discretizes the Laplacian and the source term. It is shown that only two of them provide a stable method. The accuracy of such stable methods is numerically verified on several test problems.

This paper studies a time filtered finite element method for the unsteady inductionless magnetohydrodynamic (MHD) equations. The method uses the semi-implicit backward Euler scheme with a time filter in time and adopts the standard inf-sup stable fluid pairs to discretize the velocity and pressure, and the inf-sup stable face-volume elements for solving the current density and electric potential in space. Since the time filter for the velocity is added as a separate post-processing step, the scheme can be easily incorporated into the existing backward Euler code and improves the time accuracy from first order to second order. The unique solvability, unconditional energy stability, and charge conservativeness of the scheme are also proven. In terms of the energy arguments, we establish the error estimates for the velocity, current density, and electric potential. Numerical experiments are performed to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this work, we propose and analyze a new stabilized virtual element method for the coupled Stokes-Darcy problem with Beavers-Joseph-Saffman interface condition on polygonal meshes. We derive two variants of local projection stabilization methods for the coupled Stokes-Darcy problem. The significance of local projection-based stabilization terms is that they provide reasonable control of the pressure component of the Stokes flow without involving higher-order derivative terms. The discrete inf-sup condition of the coupled Stokes-Darcy problem is established for the equal-order virtual element triplets involving velocity, hydraulic head, and pressure. The optimal error estimates are derived using the equal-order virtual elements in the energy and L 2 norms. The proposed methods have several advantages: mass conservative, avoiding the coupling of the solution components, more accessible to implement, and performing efficiently on hybrid polygonal elements. Numerical experiments are conducted to depict the flexibility of the proposed methods, validating the theoretical results. [ABSTRACT FROM AUTHOR]

In this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme's application is demonstrated through two test problems. [ABSTRACT FROM AUTHOR]

In this paper, we discuss singularly perturbed time-dependent convection–diffusion problems that arise in computational neuroscience. Specifically, we provide approaches for one-dimensional singularly perturbed parabolic partial differential difference equations (SPPPDDEs) with mixed shifts in the spatial variable using fitted operator spline in compression and adaptive spline. Temporal discretization is done by backward Euler's method, and spline methods with exponential fitting on uniform mesh are implemented in the spatial domain. For better approximations, the Richardson extrapolation technique is used, which is demonstrated by two numerical examples. The convergence of the proposed methods is investigated and found to be uniform with respect to the perturbation parameter. Graphical representations are provided to show how the shifts affect the proposed solution to the problem. [ABSTRACT FROM AUTHOR]

The time-harmonic Maxwell equations at high wavenumber k in domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly in k and an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nédélec elements of order p on a mesh with mesh size h is shown under the k-explicit scale resolution condition that (a) kh/p is sufficient small and (b) p / ln k is bounded from below. [ABSTRACT FROM AUTHOR]

The objective of this article is to develop and analyze a parameter uniform numerical method for solving 2D singularly perturbed elliptic convection-diffusion differential-difference equations. The previous studies in this area have mainly focused on cases involving small shifts, but practical scenarios may involve shifts of arbitrary sizes, both larger and smaller. Despite this, the 2D singularly perturbed elliptic differential-difference equations involving shifts of arbitrary sizes have remained unsolved. To address this challenge, a fitted operator finite difference method is developed on a special equidistant mesh that ensures the nodal point coincides with the shifts after discretization.. The scheme is rigorously analyzed to prove parameter uniform a priori error estimate in L ∞ norm. To illustrate the theoretical findings, we provide numerical results and implement numerical experiments to investigate the effect of shifts on the solution. [ABSTRACT FROM AUTHOR]

In Bonito (J. Comput. Phys. 448:110719, 2022), a local discontinous Galerkin method was proposed for approximating the large bending of prestrained plates, and in Bonito (IMA J. Numer. Anal. 43:627-662, 2023) the numerical properties of this method were explored. These works considered deformations driven predominantly by bending. Thus, a bending energy with a metric constraint was considered. We extend these results to the case of an energy with both a bending component and a nonconvex stretching component, and we also consider folding across a crease. The proposed discretization of this energy features a continuous finite element space, as well as the discrete Hessian used in Bonito (J. Comput. Phys. 448:110719, 2022; IMA J. Numer. Anal. 43:627-662, 2023). We establish the Γ -convergence of the discrete to the continuous energy and also present an energy-decreasing gradient flow for finding critical points of the discrete energy. Finally, we provide numerical simulations illustrating the convergence of minimizers and the capabilities of the model. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the Lagrange and Crouzeix–Raviart type immersed finite element methods for elliptic interface problems with nonhomogeneous interface conditions. The solution of the method is represented as a sum of two functions: one is the so-called discontinuous bubble satisfying the interface conditions approximately, and the other is the immersed finite element solution of the elliptic interface problem with homogeneous interface conditions. The discontinuous bubble can be easily constructed, since it is a piecewise linear polynomial, supported only on the triangles intersecting with the interface, determined by the nodal values or edge averages on the triangles and the given interface conditions. We prove the optimal convergence under the piecewise H 2 regularity assumption. Several numerical experiments are provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

This paper introduces a novel hybrid two-grid algorithm, combining the advantages of traditional two-level (Xu in SIAM J Sci Comput 15(1):231–237, 1994) and two-step (Huang in Appl Numer Math 118:75–86, 2017) method, for solving the steady magnetohydrodynamic system. This novel algorithm includes two steps: first, a low-order finite element pair is employed to solve the original problem on a coarse mesh grid, enhancing iteration efficiency and reducing computational time; second, a high order finite element pair is used to solve the linearized equations on a fine mesh grid, thereby increasing the order accuracy. After only one single correction step on a fine mesh grid, the hybrid two-grid algorithm achieves accuracy equivalent to that of using a high-order finite element pair. The stability analysis and error estimates for the hybrid two-grid algorithm are given. Ultimately, numerical experiments are conducted to demonstrate its efficiency and effectiveness. What’s more, the relative error is comparable to that of the two-level and two-step methods. Notably, this algorithm requires only minor modifications to the two traditional methods. [ABSTRACT FROM AUTHOR]

We investigate the application of a posteriori error estimate to a fractional optimal control problem with pointwise control constraints. Specifically, we address a problem in which the state equation is formulated as an integral form of the fractional Laplacian equation, with the control variable embedded within the state equation as a coefficient. We propose two distinct finite element discretization approaches for an optimal control problem. The first approach employs a fully discrete scheme where the control variable is discretized using piecewise constant functions. The second approach, a semi-discrete scheme, does not discretize the control variable. Using the first-order optimality condition, the second-order optimality condition, and a solution regularity analysis for the optimal control problem, we devise a posteriori error estimates. Based on the established error estimates framework, an adaptive refinement strategy is developed to help achieve the optimal convergence rate. Numerical experiments are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this manuscript, a nonlinear time-fractional diffusion equation with a generalized memory kernel is studied. Initially, the original model problem is linearized by implementing the Newton's quasilinearization technique. In the time-fractional term, a generalized Caputo derivative is considered and approximated using the non-uniform L1-scheme as the solution has a singularity at t = 0 . The main contribution of this work is to develop a generalized discrete fractional Grönwall inequality. Thereafter, permitting its use to establish the stability and analyze the error estimate, under a proper regularity condition in the L 2 -norm, and an optimal convergence order O N - (2 - ζ) is obtained for the L1-scheme with respect to the graded mesh. Numerical results are inserted to corroborate the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

The manuscript contemplates a computationally robust numerical algorithm for the solution of a singularly perturbed 2D time-delayed parabolic convection-diffusion–reaction problem involving two small parameters. The operator-splitting algorithm on a uniform mesh is used in the time direction. To handle the abrupt change in the solution due to the presence of perturbation parameters, the upwind scheme is used in the spatial direction on a layer-rectifying mesh, namely, the Shishkin mesh. This makes the accuracy in space to be one (with a logarithmic term). The use of Bakhvalov-Shishkin mesh in the spatial domain rectifies this logarithmic effect on the rate of convergence. The highly efficient Thomas algorithm is used for the computation. The devised method is claimed to be parameter uniform and globally first-order accurate. All the theoretical claims are corroborated in practice with valid numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, three-dimensional numerical algorithm is constructed to simulate the behavior of the Brinkman-Forchheimer flow and thermal fields. Numerical results of velocity, pressure and temperature are obtained by applying the efficient modified two-grid marker and cell (MAC) algorithm on staggered grids with the second-order backward difference formula (BDF2) time approximation. The modified-upwind idea is introduced to convective heat transfer equations for improving accuracy without any numerical oscillation. The second-order convergence rate can be achieved for pressure, velocity and temperature of considered three-dimensional model. Some numerical experiments are presented to illustrate the efficiency of algorithm. The numerical example with analytical solution is used to validate the effectiveness and accuracy of the algorithm by comparing with the results of traditional MAC algorithm. A time-dependent test is proposed to show a detailed sensitivity analysis to indicate the influence of parameters including the ε , Forchheimer number, Brinkman number and thermal diffusivity on the physical properties of Brinkman-Forchheimer flow and heat transfer in porous media. [ABSTRACT FROM AUTHOR]

This article examines a compact scheme employing fuzzy transform via exponential basis to solve nonlinear stationary convection–diffusion equations. The scheme executes approximated fuzzy components which estimate the solution values with fourth-order accuracy in an optimal computing time. Such an arrangement associates the approximated fuzzy components with solution values by a linear system. The Jacobian matrices in the scheme are monotone and irreducible. The proof of convergence is briefly discussed. Numerical simulations with nonlinear and linear convection–diffusion equations occurring in quantum mechanics and rheological Carreau fluid will be examined to corroborate the new scheme's utility and efficiency of computational convergence order. [ABSTRACT FROM AUTHOR]

The Scott–Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order k and a discontinuous pressure approximation of order k - 1 . It employs a "singular distance" (measured by some geometric mesh quantity Θ z ≥ 0 for triangle vertices z ) and imposes a local side condition on the pressure space associated to vertices z with Θ z = 0 . The method is inf-sup stable for any fixed regular triangulation and k ≥ 4 . However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices 0 < Θ z ≪ 1 . In this paper, we introduce a very simple parameter-dependent modification of the Scott–Vogelius element with a mesh-robust inf-sup constant. To this end, we provide sharp two-sided bounds for the inf-sup constant with an optimal dependence on the "singular distance". We characterise the critical pressures to guarantee that the effect on the divergence-free condition for the discrete velocity is negligibly small, for which we provide numerical evidence. [ABSTRACT FROM AUTHOR]

Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated volume and facet nodes, known as diagonal- E operators, are attractive for entropy-stable discretizations from an efficiency standpoint. However, there is a limited number of such operators, and those currently in existence often have a relatively high node count for a given polynomial order due to a scarcity of suitable quadrature rules. We present several new symmetric positive-weight quadrature rules on triangles and tetrahedra that are suitable for construction of diagonal- E SBP operators. For triangles, quadrature rules of degree one through twenty with facet nodes that correspond to the Legendre-Gauss-Lobatto and Legendre-Gauss quadrature rules are derived. For tetrahedra, quadrature rules of degree one through ten are presented along with the corresponding facet quadrature rules. All of the quadrature rules are provided in a supplementary data repository. The quadrature rules are used to construct novel SBP diagonal- E operators, whose accuracy and maximum time-step restrictions are studied numerically. [ABSTRACT FROM AUTHOR]

This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler's scheme for temporal discretization. With the help of Ritz–Voltera and L 2 projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of a finite volume approximation of a convective diffusive elliptic problem with Neumann boundary conditions and L 1 data. To deal with the non-coercive character of the equation and the low regularity of the right hand-side we mix the finite volume tools and the renormalized techniques. To handle the Neumann boundary conditions we choose solutions having a null median and we prove a convergence result. We present also some numerical experiments in dimension 2 to illustrate the rate of convergence. [ABSTRACT FROM AUTHOR]