Wick-stochastic finite element solution of reaction–diffusion problems

Real life reaction–diffusion problems are characterized by their inherent or externally induced uncertainties in the design parameters. This paper presents a finite element solution of reaction–diffusion equations of Wick type. Using the Wick-product properties and the Wiener–Itô chaos expansion, the stochastic variational problem is reformulated to a set of deterministic variational problems. To obtain the chaos coefficients in the corresponding deterministic reaction–diffusion, we implement the usual Galerkin finite element method using standard techniques. Once this representation is computed, the statistics of the numerical solution can be easily evaluated. Computational results are shown for one- and two-dimensional test examples.     

Convergence rate of numerical solutions to SFDEs with jumps

In this paper, we are interested in numerical solutions of stochastic functional differential equations with jumps. Under a global Lipschitz condition, we show that the pth-moment convergence of Euler-Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1/p for any p≥2. This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1/2 for any p≥2. It is therefore best to use the mean-square convergence for stochastic functional differential equations with jumps. Moreover, under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1/2, provided that local Lipschitz constants, valid on balls of radius j, do not grow faster than logj.     

Error analysis of finite element approximations of the inverse mean curvature flow arising from the general relativity

This paper proposes and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black hole theory in the general relativity. The nonlinear equation, which was derived and analyzed by Huisken and Ilmanen in (J Diff Geom 59:353–437), represents a level set formulation for the inverse mean curvature flow describing the evolution of a hypersurface whose normal velocity equals the reciprocal of its mean curvature. We first propose a finite element method for a regularized flow which involves a small parameter ɛ; a rigorous analysis is presented to study well-posedness and convergence of the scheme under certain mesh-constraints, and optimal rates of convergence are verified. We then prove uniform convergence of the finite element solution to the unique weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter ɛ both tend to zero. Computational results are provided to show the efficiency of the proposed finite element method and to numerically validate the “jumping out” phenomenon of the weak solution of the inverse mean curvature flow. Numerical studies are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter ɛ for optimal convergence of the proposed scheme. Finally, a numerical convergence study for another approach recently proposed by R. Moser (The inverse mean curvature flow and p-harmonic functions. preprint U Bath, 2005) for approximating the inverse mean curvature flow via p-harmonic functions is also included.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Spatial approximation of stochastic convolutions

We study linear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. Since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations, we propose an expansion in an arbitrary frame. We show how to obtain error estimates when the truncated expansion is used in the equation. For the stochastic heat and wave equations, we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error. Specializing the frame to biorthogonal wavelets in one variable, we show how the hierarchical structure, support and cancelation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion.     

Modified edge finite elements for photonic crystals

We consider Maxwell’s equations with periodic coefficients as it is usually done for the modeling of photonic crystals. Using Bloch/Floquet theory, the problem reduces in a standard way to a modification of the Maxwell cavity eigenproblem with periodic boundary conditions. Following [8], a modification of edge finite elements is considered for the approximation of the band gap. The method can be used with meshes of tetrahedrons or parallelepipeds. A rigorous analysis of convergence is presented, together with some preliminary numerical results in 2D, which fully confirm the robustness of the method. The analysis uses well established results on the discrete compactness for edge elements, together with new sharper interpolation estimates.     

Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.     

Block iterative solvers for higher order finite volume methods

Recently, new higher order finite volume methods (FVM) were introduced in [Z. Cai, J. Douglas, M. Park, Development and analysis of higher order finite volume methods over rectangles for elliptic equations, Adv. Comput. Math. 19 (2003) 3-33], where the linear system derived by the hybridization with Lagrange multiplier satisfying the flux consistency condition is reduced to a linear system for a pressure variable by an appropriate quadrature rule. We study the convergence of an iterative solver for this linear system. The conjugate gradient (CG) method is a natural choice to solve the system, but it seems slow, possibly due to the non-diagonal dominance of the system. In this paper, we propose block iterative methods with a reordering scheme to solve the linear system derived by the higher order FVM and prove their convergence. With a proper ordering, each block subproblem can be solved by fast methods such as the multigrid (MG) method. The numerical experiments show that these block iterative methods are much faster than CG.     

Rate of convergence of regularization procedures and finite element approximations for the total variation flow

Summary We derive rates of convergence for regularization procedures (characterized by a parameter ) and finite element approximations of the total variation flow, which arises from image processing, geometric analysis and materials sciences. Practically useful error estimates, which depend on only in low polynomial orders, are established for the proposed fully discrete finite element approximations. As a result, scaling laws which relate mesh parameters to the regularization parameter are also obtained. Numerical experiments are provided to validate the theoretical results and show efficiency of the proposed numerical methods.     

Finite element approximation of a periodic Ginzburg-Landau model for type-II superconductors

Summary We consider efficient finite element algorithms for the computational simulation of type-II superconductors. The algorithms are based on discretizations of a periodic Ginzburg-Landau model. Periodicity is defined with respect to a non-orthogonal lattice that is not necessarily aligned with the coordinate axes; also, the primary dependent variables employed in the model satisfy non-standard quasi-periodic boundary conditions. After introducing the model, we define finite element schemes, derive error estimates of optimal order, and present the results of some numerical calculations. For a similar quality of simulation, the resulting algorithms seem to be significantly less costly than are previously used numerical approximation methods.     

Controllability method for acoustic scattering with spectral elements

We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improvements in efficiency due to the higher order spectral elements. For a given accuracy, the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method. In addition, by using higher order polynomial basis the influence of the pollution effect is reduced.     

Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.     

Split-step forward methods for stochastic differential equations

In this paper we discuss split-step forward methods for solving Itô stochastic differential equations (SDEs). Eight fully explicit methods, the drifting split-step Euler (DRSSE) method, the diffused split-step Euler (DISSE) method and the three-stage Milstein (TSM 1a-TSM 1f) methods, are constructed based on Euler-Maruyama method and Milstein method, respectively, in this paper. Their order of strong convergence is proved. The analysis of stability shows that the mean-square stability properties of the methods derived in this paper are improved on the original methods. The numerical results show the effectiveness of these methods in the pathwise approximation of Itô SDEs.     

Preserving exponential mean-square stability in the simulation of hybrid stochastic differential equations

Positive results are derived concerning the long time dynamics of fixed step size numerical simulations of stochastic differential equation systems with Markovian switching. Euler–Maruyama and implicit theta-method discretisations are shown to capture exponential mean-square stability for all sufficiently small time-steps under appropriate conditions. Moreover, the decay rate, as measured by the second moment Lyapunov exponent, can be reproduced arbitrarily accurately. New finite-time convergence results are derived as an intermediate step in this analysis. We also show, however, that the mean-square A-stability of the theta method does not carry through to this switching scenario. The proof techniques are quite general and hence have the potential to be applied to other numerical methods.     

基于嵌入式多项式混沌展开法的随机边界下流动与传热问题不确定性量化

提出了一种嵌入式多项式混沌展开(polynomial chaos expansion, PCE)的随机边界条件下流动与传热问题不确定性量化方法及有限元程序框架.该方法利用Karhunen-Loeve展开表达随机输入边界条件,以及嵌入式多项式混沌展开法表达输出随机场;同时利用谱分解技术将控制方程转化为一组确定性控制方程,并对每个多项式混沌进行求解得到其统计特征.与Monte-Carlo法相比,该方法能够准确高效地预测随机边界条件下流动与传热问题的不确定性特征,同时可以节省大量计算资源.     

A parallel four step domain decomposition scheme for coupled forward–backward stochastic differential equations

Motivated by the idea of imposing paralleling computing on solving stochastic differential equations (SDEs), we introduce a new domain decomposition scheme to solve forward–backward stochastic differential equations (FBSDEs) parallel. We reconstruct the four step scheme in Ma et al. (1994) [1] and then associate it with the idea of domain decomposition methods. We also introduce a new technique to prove the convergence of domain decomposition methods for systems of quasilinear parabolic equations and use it to prove the convergence of our scheme for the FBSDEs.     

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.     

Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems

We generalise the current theory of optimal strong convergence rates for implicit Euler-based methods by allowing for Poisson-driven jumps in a stochastic differential equation (SDE). More precisely, we show that under one-sided Lipschitz and polynomial growth conditions on the drift coefficient and global Lipschitz conditions on the diffusion and jump coefficients, three variants of backward Euler converge with strong order of one half. The analysis exploits a relation between the backward and explicit Euler methods.     

Superconvergence of new mixed finite element spaces

In this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in [ETNA, Vol. 37, pp. 189-201, 2010]. Among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids, this space has the smallest unknowns. However, the scalar variable is only suboptimal in general; thus we have employed a post-processing technique for the scalar variable. As a byproduct, we have obtained a superconvergence on a rectangular grid. The superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory, but that of a scalar variable requires a new technique, especially for k=1. Numerical experiments are provided to support the theory.