We consider numerical methods to solve the Allen-Cahn equation using the secondorder Crank-Nicolson scheme in time and the second-order central difference approach in space. The existence of the finite difference solution is proved with the help of Browder fixed point theorem. The difference scheme is showed to be unconditionally convergent in L_∞ norm by constructing an auxiliary Lipschitz continuous function. Based on this result, it is demonstrated that the difference scheme preserves the maximum principle without any restrictions on spatial step size and temporal step size. The numerical experiments also verify the reliability of the method.

In this paper, we study a nonlinear first-order singularly perturbed Volterra integrodifferential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.

We propose a θ-L approach for solving a sharp-interface model about simulating solidstate dewetting of thin films with isotropic/weakly anisotropic surface energies. The sharpinterface model is governed by surface diffusion and contact line migration. For solving the model, traditional numerical methods usually suffer from the severe stability constraint and/or the mesh distribution trouble. In the θ-L approach, we introduce a useful tangential velocity along the evolving interface and utilize a new set of variables (i.e., the tangential angle θ and the total length L of the interface curve), so that it not only could reduce the stiffness resulted from the surface tension, but also could ensure the mesh equidistribution property during the evolution. Furthermore, it can achieve second-order accuracy when implemented by a semi-implicit linear finite element method. Numerical results are reported to demonstrate that the proposed θ-L approach is efficient and accurate.

This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the L^∞(0,T;L²(Ω))-norm. Finally, a numerical experiment is performed to illustrate the performance of the derived estimators.

Implicit-explicit Runge-Kutta-Rosenbrock methods are proposed to solve nonlinear stiff ordinary differential equations by combining linearly implicit Rosenbrock methods with explicit Runge-Kutta methods. First, the general order conditions up to order 3 are obtained. Then, for the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, the corresponding errors of the implicit-explicit methods are analysed. At last, some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the methods.

In this paper, we consider the knot placement problem in B-spline curve approximation. A novel two-stage framework is proposed for addressing this problem. In the first step, the l_∞,1-norm model is introduced for the sparse selection of candidate knots from an initial knot vector. By this step, the knot number is determined. In the second step, knot positions are formulated into a nonlinear optimization problem and optimized by a global optimization algorithm — the differential evolution algorithm (DE). The candidate knots selected in the first step are served for initial values of the DE algorithm. Since the candidate knots provide a good guess of knot positions, the DE algorithm can quickly converge. One advantage of the proposed algorithm is that the knot number and knot positions are determined automatically. Compared with the current existing algorithms, the proposed algorithm finds approximations with smaller fitting error when the knot number is fixed in advance. Furthermore, the proposed algorithm is robust to noisy data and can handle with few data points. We illustrate with some examples and applications.

We present a stochastic trust-region model-based framework in which its radius is related to the probabilistic models. Especially, we propose a specific algorithm termed STRME, in which the trust-region radius depends linearly on the gradient used to define the latest model. The complexity results of the STRME method in nonconvex, convex and strongly convex settings are presented, which match those of the existing algorithms based on probabilistic properties. In addition, several numerical experiments are carried out to reveal the benefits of the proposed methods compared to the existing stochastic trust-region methods and other relevant stochastic gradient methods.

This paper focuses on a fast and high-order finite difference method for two-dimensional space-fractional complex Ginzburg-Landau equations. We firstly establish a three-level finite difference scheme for the time variable followed by the linearized technique of the nonlinear term. Then the fourth-order compact finite difference method is employed to discretize the spatial variables. Hence the accuracy of the discretization is $\mathcal{O}$(τ² + $h_1^4$ + $h_2^4$) in L₂-norm, where τ is the temporal step-size, both h₁ and h₂ denote spatial mesh sizes in x- and y- directions, respectively. The rigorous theoretical analysis, including the uniqueness, the almost unconditional stability, and the convergence, is studied via the energy argument. Practically, the discretized system holds the block Toeplitz structure. Therefore, the coefficient Toeplitz-like matrix only requires $\mathcal{O}$(M₁M₂) memory storage, and the matrix-vector multiplication can be carried out in $\mathcal{O}$(M₁M₂(log M₁ + log M₂)) computational complexity by the fast Fourier transformation, where M₁ and M₂ denote the numbers of the spatial grids in two different directions. In order to solve the resulting Toeplitz-like system quickly, an efficient preconditioner with the Krylov subspace method is proposed to speed up the iteration rate. Numerical results are given to demonstrate the well performance of the proposed method.

This paper deals with discontinuous dual reciprocity boundary element method for solving an inverse source problem. The aim of this work is to determine the source term in elliptic equations for nonhomogenous anisotropic media, where some additional boundary measurements are required. An equivalent formulation to the primary inverse problem is established based on the minimization of a functional cost, where a regularization term is employed to eliminate the oscillations of the noisy data. Moreover, an efficient algorithm is presented and tested for some numerical examples.

A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied. The scheme is constructed with two-layer coupled discretization (TLCD) at each time step. It does not stir numerical oscillation, while permits large time step length, and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes, the Crank-Nicolson (CN) scheme and the backward difference formula second-order (BDF2) scheme. By developing a new reasoning technique, we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers, and prove rigorously the TLCD scheme is uniquely solvable, unconditionally stable, and has second-order convergence in both space and time. Numerical tests verify the theoretical results, and illustrate its superiority over the CN and BDF2 schemes.

We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane. This new class of finite elements, which is called composite finite elements, was first introduced by Hackbusch and Sauter [Numer. Math., 75 (1997), pp. 447-472] for the approximation of partial differential equations on domains with complicated geometry. The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving partial differential equations by domain discretization method. The composite finite element method introduces two-scale grid for discretization of the domain, the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only. While the fine-scale grid is used to resolve the Dirichlet boundary condition, the dimension of the finite element space depends only on the coarse-scale grid. As a consequence, the resulting linear system will have a fewer number of unknowns. A continuous, piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods. We have derived the error estimates in the L^∞(L²)-norm for both semidiscrete and fully discrete schemes. Moreover, numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.

Precision matrix estimation is an important problem in statistical data analysis. This paper proposes a sparse precision matrix estimation approach, based on CLIME estimator and an efficient algorithm GISS^ρ that was originally proposed for l₁ sparse signal recovery in compressed sensing. The asymptotic convergence rate for sparse precision matrix estimation is analyzed with respect to the new stopping criteria of the proposed GISS^ρ algorithm. Finally, numerical comparison of GISS^ρ with other sparse recovery algorithms, such as ADMM and HTP in three settings of precision matrix estimation is provided and the numerical results show the advantages of the proposed algorithm.

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order 1/2 and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.

In this work we introduce two new Barzilai and Borwein-like steps sizes for the classical gradient method for strictly convex quadratic optimization problems. The proposed step sizes employ second-order information in order to obtain faster gradient-type methods. Both step sizes are derived from two unconstrained optimization models that involve approximate information of the Hessian of the objective function. A convergence analysis of the proposed algorithm is provided. Some numerical experiments are performed in order to compare the efficiency and effectiveness of the proposed methods with similar methods in the literature. Experimentally, it is observed that our proposals accelerate the gradient method at nearly no extra computational cost, which makes our proposal a good alternative to solve large-scale problems.

A combined scheme of the improved two-grid technique with the block-centered finite difference method is constructed and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and two linear problems based on the coarse-grid solutions and one Newton iteration is considered on a fine grid of size h. We provide the rigorous error estimate, which demonstrates that our scheme converges with order O(Δt^2-α+h²+H⁴) on non-uniform rectangular grid. This result indicates that the improved two-grid method can obtain asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H²). Finally, numerical tests confirm the theoretical results of the presented method.

For symmetric tensors, computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere. In this paper, we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors. One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance. The other one is that a projection scheme is used to ensure the feasibility of all iteratives. Global convergence and local quadratic convergence of our algorithm are established, respectively. The preliminary numerical results show the efficiency of the proposed algorithm.

Shifted symmetric higher-order power method (SS-HOPM) has been proved effective in solving the nonlinear eigenvalue problem oriented from the Bose-Einstein Condensation (BEC-like NEP for short) both theoretically and numerically. However, the convergence of the sequence generated by SS-HOPM is based on the assumption that the real eigenpairs of BEC-like NEP are finite. In this paper, we will establish the point-wise convergence via Lojasiewicz inequality by introducing a new related sequence.

In this paper, we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time. A novel regularization method, which we call the exponential Tikhonov regularization method with a parameter γ, is proposed to solve the inverse source problem, and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules. When γ is less than or equal to zero, the optimal convergence rate can be achieved and it is independent of the value of γ. However, when γ is great than zero, the optimal convergence rate depends on the value of γ which is related to the regularity of the unknown source. Finally, numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.

We consider optimal shape design in Stokes flow using H¹ shape gradient flows based on the distributed Eulerian derivatives. MINI element is used for discretizations of Stokes equation and Galerkin finite element is used for discretizations of distributed and boundary H¹ shape gradient flows. Convergence analysis with a priori error estimates is provided under general and different regularity assumptions. We investigate the performances of shape gradient descent algorithms for energy dissipation minimization and obstacle flow. Numerical comparisons in 2D and 3D show that the distributed H¹ shape gradient flow is more accurate than the popular boundary type. The corresponding distributed shape gradient algorithm is more effective.

The purpose of this paper is to discuss representations of high order C⁰ finite element spaces on simplicial meshes in any dimension. When computing with high order piecewise polynomials the conditioning of the basis is likely to be important. The main result of this paper is a construction of representations by frames such that the associated L² condition number is bounded independently of the polynomial degree. To our knowledge, such a representation has not been presented earlier. The main tools we will use for the construction is the bubble transform, introduced previously in[1], and properties of Jacobi polynomials on simplexes in higher dimensions. We also include a brief discussion of preconditioned iterative methods for the finite element systems in the setting of representations by frames.

In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by H¹-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice h = H ².

The method of data-driven tight frame has been shown very useful in image restoration problems. We consider in this paper extending this important technique, by incorporating L₁ data fidelity into the original data-driven model, for removing impulsive noise which is a very common and basic type of noise in image data. The model contains three variables and can be solved through an efficient iterative alternating minimization algorithm in patch implementation, where the tight frame is dynamically updated. It constructs a tight frame system from the input corrupted image adaptively, and then removes impulsive noise by the derived system. We also show that the sequence generated by our algorithm converges globally to a stationary point of the optimization model. Numerical experiments and comparisons demonstrate that our approach performs well for various kinds of images. This benefits from its data-driven nature and the learned tight frames from input images capture richer image structures adaptively.

In this paper, ETD3-Padé and ETD4-Padé Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Padé approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Padé schemes. An unconditional L² numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of ^O(k³ + h^r) (ETD3- Padé) or O(k⁴ + h^r) (ETD4-Padé) in the L² norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

This paper presents a second-order direct arbitrary Lagrangian Eulerian (ALE) method for compressible flow in two-dimensional cylindrical geometry. This algorithm has half-face fluxes and a nodal velocity solver, which can ensure the compatibility between edge fluxes and the nodal flow intrinsically. In two-dimensional cylindrical geometry, the control volume scheme and the area-weighted scheme are used respectively, which are distinguished by the discretizations for the source term in the momentum equation. The two-dimensional second-order extensions of these schemes are constructed by employing the monotone upwind scheme of conservation law (MUSCL) on unstructured meshes. Numerical results are provided to assess the robustness and accuracy of these new schemes.

In this paper, we present a finite element algorithm for the time-dependent nematic liquid crystal flow based on the Gauge-Uzawa method. This algorithm combines the Gauge and Uzawa methods within a finite element variational formulation, which is a fully discrete projection type algorithm, whereas many projection methods have been studied without space discretization. Besides, error estimates for velocity and molecular orientation of the nematic liquid crystal flow are shown. Finally, numerical results are given to show that the presented algorithm is reliable and confirm the theoretical analysis.

We propose a novel algorithm, based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara, Camassa-Holm and Benjamin-Ono equations. The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error. We present several numerical experiments to demonstrate that PINNs can approximate solutions of these dispersive PDEs very accurately.

In this paper, the monolithic multigrid method is investigated for reduced magnetohydrodynamic equations. We propose a diagonal Braess-Sarazin smoother for the finite element discrete system and prove the uniform convergence of the MMG method with respect to mesh sizes. A multigrid-preconditioned FGMRES method is proposed to solve the magnetohydrodynamic equations. It turns out to be robust for relatively large physical parameters. By extensive numerical experiments, we demonstrate the optimality of the monolithic multigrid method with respect to the number of degrees of freedom.

In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the L²-norm for even degree polynomial approximations. On uniform meshes, the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions, improving previous results. The theoretical findings are found to be sharp and consistent with numerical results.

We analyze the convergence of the weighted nonlocal Laplacian (WNLL) on the high dimensional randomly distributed point cloud. Our analysis reveals the importance of the scaling weight, μ~|P|/|S|with|P|and|S|being the number of entire and labeled data, respectively, in WNLL. The established result gives a theoretical foundation of the WNLL for high dimensional data interpolation.

Randomize-then-optimize (RTO) is widely used for sampling from posterior distributions in Bayesian inverse problems. However, RTO can be computationally intensive for complexity problems due to repetitive evaluations of the expensive forward model and its gradient. In this work, we present a novel goal-oriented deep neural networks (DNN) surrogate approach to substantially reduce the computation burden of RTO. In particular, we propose to drawn the training points for the DNN-surrogate from a local approximated posterior distribution-yielding a flexible and efficient sampling algorithm that converges to the direct RTO approach. We present a Bayesian inverse problem governed by elliptic PDEs to demonstrate the computational accuracy and efficiency of our DNN-RTO approach, which shows that DNN-RTO can significantly outperform the traditional RTO.

In this work we introduce and analyze a mixed virtual element method (mixed-VEM) for the two-dimensional stationary Boussinesq problem. The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity, which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor, the velocity and the temperature, whereas the pressure is computed via a postprocessing formula. In addition, an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation. Regarding the discrete problem, we follow the approach employed in a previous work dealing with the Navier-Stokes equations, and couple it with a VEM for the convection-diffusion equation modelling the temperature. More precisely, we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of ${\Bbb H}$(div) and H¹, respectively, whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H¹. In this way, we make use of the L²-orthogonal projectors onto suitable polynomial spaces, which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations. On the other hand, in order to manipulate the bilinear form associated to the heat equations, we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor, which represents the thermal conductivity, is variable. Next, the corresponding solvability analysis is performed using again appropriate fixed-point arguments. Further, Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure. The corresponding rates of convergence are also established. Finally, several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.

Iterative ILU factorizations are constructed, analyzed and applied as preconditioners to solve both linear systems and eigenproblems. The computational kernels of these novel Iterative ILU factorizations are sparse matrix-matrix multiplications, which are easy and efficient to implement on both serial and parallel computer architectures and can take full advantage of existing matrix-matrix multiplication codes. We also introduce level-based and threshold-based algorithms in order to enhance the accuracy of the proposed Iterative ILU factorizations. The results of several numerical experiments illustrate the efficiency of the proposed preconditioners to solve both linear systems and eigenvalue problems.

In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order α ∈ (1, 2) and α₁ ∈ (0, 1). Numerical stability and optimal error estimate O(h^r+1 + H^2r+2 + τ^{min-3-α,2-α1}}) in L²-norm are presented for two-grid scheme, where t, H and h are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.

In this paper, we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems. We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method, though they have essentially different bilinear forms. More precisely, we prove its reliability and efficiency for the actual error measured in the standard DG norm. We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution. Numerical results are presented to verify the theoretical analysis.

The present paper conjectures a topological condition which classifies a T-spline into standard, semi-standard and non-standard. We also provide the basic framework to prove the conjecture on the classification of semi-standard T-splines and give the proof for the semi-standard of bi-degree (1, d) and (d, 1) T-splines.

In this paper, we present a unified finite volume method preserving discrete maximum principle (DMP) for the conjugate heat transfer problems with general interface conditions. We prove the existence of the numerical solution and the DMP-preserving property. Numerical experiments show that the nonlinear iteration numbers of the scheme in [24] increase rapidly when the interfacial coefficients decrease to zero. In contrast, the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero, which reveals that the unified scheme is more robust than the scheme in [24]. The accuracy and DMP-preserving property of the scheme are also verified in the numerical experiments.

In this paper, we consider a modified alternating positive semidefinite splitting preconditioner for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current model. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied for both simple and general topology. Numerical results demonstrate the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz condition. These assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations (SODEs) under “minimum assumptions” were studied. As a result, the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear SODEs. There are several difficulties which need to be overcome for this generalization. First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty. It turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix. In this paper we use a finite element interpolation technique to discretize the nonlinear drift term. Second, in order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the H¹-seminorm of the numerical solution must be established, which are difficult and delicate. A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the goal. Finally, stability estimates for the second and higher order moments of the L²-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant. This is done by utilizing the interpolation theory and the higher moment estimates for the H¹-seminorm of the numerical solution. After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence. Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.

Numerical integration poses greater challenges in Galerkin meshless methods than finite element methods owing to the non-polynomial feature of meshless shape functions. The reproducing kernel gradient smoothing integration (RKGSI) is one of the optimal numerical integration techniques in Galerkin meshless methods with minimum integration points. In this paper, properties, quadrature rules and the effect of the RKGSI on meshless methods are analyzed. The existence, uniqueness and error estimates of the solution of Galerkin meshless methods under numerical integration with the RKGSI are established. A procedure on how to choose quadrature rules to recover the optimal convergence rate is presented.

Schwarz method is put forward to solve second order backward stochastic differential equations (2BSDEs) in this work. We will analyze uniqueness, convergence, stability and optimality of the proposed method. Moreover, several simulation results are presented to demonstrate the effectiveness; several applications of the 2BSDEs are investigated. It is concluded from these results that the proposed the method is powerful to calculate the 2BSDEs listing from the financial engineering.