Abstract: We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods. Numerical experiments verify the theory and validate the performance of the proposed method.

Abstract: In this paper we develop an evolution of the $C^1$ virtual elements of minimal degree for the approximation of the Cahn--Hilliard equation. The proposed method has the advantage of being conforming in $H^2$ and making use of a very simple set of degrees of freedom, namely, 3 degrees of freedom per vertex of the mesh. Moreover, although the present method is new also on triangles, it can make use of general polygonal meshes. As a theoretical and practical support, we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests.

Abstract: Similar to other numerical methods developed for the simulation of fluid flow, the finite volume method transforms the set of partial differential equations into a system of linear algebraic equations. Nevertheless, the discretization procedure used in the finite volume method is distinctive and involves two basic steps. In the first step, the partial differential equations are integrated and transformed into balance equations over an element. This involves changing the surface and volume integrals into discrete algebraic relations over elements and their surfaces using an integration quadrature of a specified order of accuracy. The result is a set of semi-discretized equations. In the second step, interpolation profiles are chosen to approximate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations. The current chapter details the first discretization step and presents a broad review of numerical issues pertaining to the finite volume method. This provides a solid foundation on which to expand in the coming chapters where the focus will be on the discretization of the various parts of the general conservation equation. In both steps, the selected approximations affect the accuracy and robustness of the resulting numerics. It is therefore important to define some guiding principles for informing the selection process.

Abstract: In this paper, we consider a numerical method for the time-fractional diffusion equation. The method uses a high order finite difference method to approximate the fractional derivative in time, resulting in a time stepping scheme for the underlying equation. Then the resulting equation is discretized in space by using a spectral method based on the Legendre polynomials. The main body of this paper is devoted to carry out a rigorous analysis for the stability and convergence of the time stepping scheme. As a by-product and direct extension of our previous work, an error estimate for the spatial discretization is also provided. The key contribution of the paper is the proof of the ($3-\alpha$)-order convergence of the time scheme, where $\alpha$ is the order of the time-fractional derivative. Then the theoretical result is validated by a number of numerical tests. To the best of our knowledge, this is the first proof for the stability of the ($3-\alpha$)-order scheme for the time-fractional diffusion equation.

Abstract: Scaling algorithms for entropic transport-type problems have become a very popular numerical method, encompassing Wasserstein barycenters, multi-marginal problems, gradient flows and unbalanced transport. However, a standard implementation of the scaling algorithm has several numerical limitations: the scaling factors diverge and convergence becomes impractically slow as the entropy regularization approaches zero. Moreover, handling the dense kernel matrix becomes unfeasible for large problems. To address this, we combine several modifications: A log-domain stabilized formulation, the well-known epsilon-scaling heuristic, an adaptive truncation of the kernel and a coarse-to-fine scheme. This permits the solution of larger problems with smaller regularization and negligible truncation error. A new convergence analysis of the Sinkhorn algorithm is developed, working towards a better understanding of epsilon-scaling. Numerical examples illustrate efficiency and versatility of the modified algorithm.

Abstract: This review paper presents a methodological study on possible and existing meshfree methods for solving the partial differential equations (PDEs) governing solid mechanics problems, based mainly on the research work in the past two decades at the authors group. We start with a discussion on the general steps in a meshfree method based on nodes, with the displacements as the primary variables. We then examine the major techniques used in each of these steps: (1) techniques for displacement function approximations using nodes, (2) approximation of the gradient of the displacements or strains based on nodes and a background T-cells that can be automatically generated and refined, and (3) formulation techniques for producing algebraic equations. The function approximation techniques include node-based interpolation methods, cell-based interpolation methods, function smoothing techniques, and moving least squares approximation techniques. The gradient approximation includes direct differentiation, gradient smoothing, and special strain construction. Formulation techniques include strong-form, weakform, local weakform, weak-strong-form, and weakened weakform (W2). In theory, a meshfree method can be developed using a combination of function approximation, gradient approximation, and formulation techniques, which can lead to matrix of a large number of possible methods. This review attempts to provide an overall methodological review, rather than a usual review of comparing different methods. We hope to show readers the differences between the forests, and just between the trees.

Abstract: It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.

Abstract: A new optimized two-step hybrid block method for the numerical integration of general second-order initial value problems is presented. The method considers two intra-step points which are selected adequately in order to optimize the local truncation errors of the main formulas for the solution and the derivative at the final point of the block. The new method is zero-stable and consistent with fifth algebraic order. Numerical experiments used revealed the superiority of the new method for solving this kind of problems, in comparison with methods of similar characteristics in the literature.

Abstract: We propose a numerical method for solving general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on nonuniform grids. The properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant set containing the initial data is an invariant domain for the method. The invariant domain property is proved for any hyperbolic system provided a CFL condition holds. The solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system. The method is formally first-order accurate in space and can be made high-order in time by using strong stability preserving algorithms. This technique extends to continuous finite elements the work of [D. Hoff, Math. Comp., 33 (1979), pp. 1171--1193], [D. Hoff, Trans. Amer. Math. Soc., 289 (1985), pp. 591--610], and [H. Frid, Arch. Ration. Mech. Anal., 160 (2001), pp. 245--269].

Abstract: In this paper, new operational matrices for shifted Legendre orthonormal polynomial are derived. This polynomial is used as a basis function for developing a new numerical technique for the delay fractional optimal control problem. The fractional integral is described in the Riemann---Liouville sense, while the fractional derivative is described in the Caputo sense. The operational matrix of fractional integrals is used together with the Lagrange multiplier method for the constrained extremum in order to minimize the performance index. The problem is then reduced to a problem consists of a system of easily solvable algebraic equations. Three numerical examples of different types of delay fractional optimal control problems are implemented with their approximate solutions for confirming the high accuracy and applicability of the proposed method.

Abstract: A new generalized model of two-temperature thermoelasticity theory with time-delay and Kernel function is constructed. Taylor theorem in terms of memory-dependent derivatives is proved. The governing coupled equations of the new generalized thermoelasticity with time-delay and Kernel function, which can be chosen freely according to the necessity of applications, are applied to a one-dimensional problem of a half-space. The bounding surface is taken to be traction free and subjected to a time-dependent thermal shock. Laplace transforms technique will be used to obtain the general solution in a closed form. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and its graphs, conclusions about the new theory have been constructed. Some comparisons are shown in the figures to estimate the effects of the temperature discrepancy and time-delay parameter on all of the studied fields.

Abstract: This paper presents a new numerical method for solving fractional optimal control problems (FOCPs). The fractional derivative in the dynamic system is described in the Caputo sense. The method is based upon Bernoulli polynomials. The operational matrices of fractional Riemann–Liouville integration and multiplication for Bernoulli polynomials are derived. The error upper bound for the operational matrix of the fractional integration is also given. The properties of Bernoulli polynomials are utilized to reduce the given optimization problems to the system of algebraic equations. By using Newton’s iterative method, this system is solved and the solution of FOCPs are achieved. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Abstract: Frictional sliding and crack growth are two main dissipation processes in quasi brittle materials. The frictional sliding along closed cracks is the origin of macroscopic plastic deformation while the crack growth induces a material damage. The main difficulty of modeling is to consider the inherent coupling between these two processes. Various models and associated numerical algorithms have been proposed. But there are so far no analytical solutions even for simple loading paths for the validation of such algorithms. In this paper, we first present a micro-mechanical model taking into account the damage-friction coupling for a large class of quasi brittle materials. The model is formulated by combining a linear homogenization procedure with the Mori–Tanaka scheme and the irreversible thermodynamics framework. As an original contribution, a series of analytical solutions of stress–strain relations are developed for various loading paths. Based on the micro-mechanical model, two numerical integration algorithms are exploited. The first one involves a coupled friction/damage correction scheme, which is consistent with the coupling nature of the constitutive model. The second one contains a friction/damage decoupling scheme with two consecutive steps: the friction correction followed by the damage correction. With the analytical solutions as reference results, the two algorithms are assessed through a series of numerical tests. It is found that the decoupling correction scheme is efficient to guarantee a systematic numerical convergence.

Abstract: This article adapts an operational matrix formulation of the collocation method for the one- and two-dimensional nonlinear fractional sub-diffusion equations (FSDEs). In the proposed collocation approach, the double and triple shifted Jacobi polynomials are used as base functions for approximate solutions of the one- and two-dimensional cases. The space and time fractional derivatives given in the underline problems are expressed by means of Jacobi operational matrices. This investigates spectral collocation schemes for both temporal and spatial discretizations. Thereby, the expansion coefficients are then determined by reducing the FSDEs, with their initial and boundary conditions, into systems of nonlinear algebraic equations which are far easier to be solved. Furthermore, the error of the approximate solution is estimated theoretically along with graphical analysis to confirm the exponential convergence rate of the proposed method in both spatial and temporal discretizations. In order to show the high accuracy of our algorithms, we report the numerical results of some numerical examples and compare our numerical results with those reported in the literature.

Abstract: This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg---Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (SIAM J Numer Anal 40(3):1041---1063, 2002) and a newly proposed explicit variant of the Euler---Maruyama scheme, the so called projected Euler---Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments.

Abstract: In this paper we discuss a new discretization for the Biot equations. The discretization treats the coupled system of deformation and flow directly, as opposed to combining discretizations for the two separate subproblems. The coupled discretization has the following key properties, the combination of which is novel: (1) The variables for the pressure and displacement are co-located and are as sparse as possible (e.g., one displacement vector and one scalar pressure per cell center). (2) With locally computable restrictions on grid types, the discretization is stable with respect to the limits of incompressible fluid and small time-steps. (3) No artificial stabilization term has been introduced. Furthermore, due to the finite volume structure embedded in the discretization, explicit local expressions for both momentum-balancing forces and mass-conservative fluid fluxes are available. We prove stability of the proposed method with respect to all relevant limits. Together with consistency, this proves conve...