Weak Galerkin finite element methods for linear parabolic integro‐differential equations

The semidiscrete and fully discrete weak Galerkin finite element schemes for the linear parabolic integro‐differential equations are proposed. Optimal order error estimates are established for the corresponding numerical approximations in both and norms. Numerical experiments illustrating the error behaviors are provided.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1357–1377, 2016     

Minimum time‐step criteria for the Galerkin finite element methods applied to one‐dimensional parabolic partial differential equations

The finite element method has been well established for numerically solving parabolic partial differential equations (PDEs). Also it is well known that a too large time step should not be chosen in order to obtain a stable and accurate numerical solution. In this article, accuracy analysis shows that a too small time step should not be chosen either for some time‐stepping schemes. Otherwise, the accuracy of the numerical solution cannot be improved or can even be worsened in some cases. Furthermore, the so‐called minimum time step criteria are established for the Crank‐Nicolson scheme, the Galerkin‐time scheme, and the backward‐difference scheme used in the temporal discretization. For the forward‐difference scheme, no minimum time step exists as far as the accuracy is concerned. In the accuracy analysis, no specific initial and boundary conditions are invoked so that such established criteria can be applied to the parabolic PDEs subject to any initial and boundary conditions. These minimum time step criteria are verified in a series of numerical experiments for a one‐dimensional transient field problem with a known analytical solution. The minimum time step criteria developed in this study are useful for choosing appropriate time steps in numerical simulations of practical engineering problems. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp-DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.     

Weak Galerkin finite element methods for Parabolic equations

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal‐order error estimates in both H¹ and L² norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal -th order of accuracy when using piecewise -th degree polynomials, under the condition that is greater than or equal to the order of the equation.

     

On discontinuous Galerkin finite element method for singularly perturbed delay differential equations

A nonsymmetric discontinuous Galerkin FEM with interior penalties has been applied to one-dimensional singularly perturbed problem with a constant negative shift. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Numerical experiments support theoretical findings.     

Analysis of finite element approximation and quadrature of Volterra integral equations

In this article we study Galerkin finite element approximations to integral equations of the Volterra type. Our prime concern is the noncoercive case, which is not covered by the standard finite element theory. The question of rates of convergence is studied for the case where an exact stiffness matrix is available, as well as the case where the latter is approximated via quadrature rules. The optimality of these rules is also considered from the point of view of the effect the choice of the quadrature has on the overall rate of convergence. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 663–672, 1997     

Preconditioning discretizations of systems of partial differential equations

This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the summer of 2008. The main focus will be on an abstract approach to the construction of preconditioners for symmetric linear systems in a Hilbert space setting. Typical examples that are covered by this theory are systems of partial differential equations which correspond to saddle point problems. We will argue that the mapping properties of the coefficient operators suggest that block diagonal preconditioners are natural choices for these systems. To illustrate our approach a number of examples will be considered. In particular, parameter‐dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several models which have previously been discussed in the literature. However, here each example is discussed with reference to a more unified abstract approach. Copyright © 2010 John Wiley & Sons, Ltd.     

On the approximation of stochastic partial differential equations i

The stability of abstract stochastic partial differential equations with respect to the simultaneous perturbation of the driving processes and of the differential operators is investigated. The results obtained here will be applied to concrete stochastic partial differential equations in the continuation of this paper     

Liouville-Green asymptotic approximation for a class of matrix differential equations and semi-discretized partial differential equations

A Liouville-Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y^″=[f(t)A+G(t)]Y on [a,+∞), where A and G(t) are matrices and f(t) is scalar. This includes the case of an “asymptotically constant” (not necessarily diagonalizable) coefficient A (when f(t)≡1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown. Several examples, some concerning semi-discretized wave and convection-diffusion equations, are given.     

Newton iteration for partial differential equations and the approximation of the identity

It is known that the critical condition which guarantees quadratic convergence of approximate Newton methods is an approximation of the identity condition. This requires that the composition of the numerical inversion of the Fréchet derivative with the derivative itself approximate the identity to an accuracy calibrated by the residual. For example, the celebrated quadratic convergence theorem of Kantorovich can be proven when this holds, subject to regularity and stability of the derivative map. In this paper, we study what happens when this condition is not evident a priori but is observed a posteriori. Through an in-depth example involving a semilinear elliptic boundary value problem, and some general theory, we study the condition in the context of dual norms, and the effect upon convergence. We also discuss the connection to Nash iteration.     

Alternating direction finite volume element methods for 2D parabolic partial differential equations

On the basis of rectangular partition and bilinear interpolation, this article presents alternating direction finite volume element methods for two dimensional parabolic partial differential equations and gives three computational schemes, one is analogous to Douglas finite difference scheme with second order splitting error, the second has third order splitting error, and the third is an extended locally one dimensional scheme. Optimal L² norm or H¹ semi‐norm error estimates are obtained for these schemes. Finally, two numerical examples illustrate the effectiveness of the schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

H1‐Galerkin expanded mixed finite element methods for nonlinear pseudo‐parabolic integro‐differential equations

H¹‐Galerkin mixed finite element method combined with expanded mixed element method is discussed for nonlinear pseudo‐parabolic integro‐differential equations. We conduct theoretical analysis to study the existence and uniqueness of numerical solutions to the discrete scheme. A priori error estimates are derived for the unknown function, gradient function, and flux. Numerical example is presented to illustrate the effectiveness of the proposed scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Fitted Galerkin spectral method to solve delay partial differential equations

In this paper, we consider a class of parabolic partial differential equations with a time delay. The first model equation is the mixed problems for scalar generalized diffusion equation with a delay, whereas the second model equation is a delayed reaction‐diffusion equation. Both of these models have inherent complex nature because of which their analytical solutions are hardly obtainable, and therefore, one has to seek numerical treatments for their approximate solutions. To this end, we develop a fitted Galerkin spectral method for solving this problem. We derive optimal error estimates based on weak formulations for the fully discrete problems. Some numerical experiments are also provided at the end. Copyright © 2015 John Wiley & Sons, Ltd.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

A numerical approximation of parabolic stochastic partial differential equations driven by a Poisson random measure

The paper deals with SPDEs driven by Poisson random measures in Banach spaces and its numerical approximation. We investigate the accuracy of space and time approximation. As the space approximation we consider spectral methods and as time approximation the implicit Euler scheme and the explicit Euler scheme. AMS subject classification (2000) 60H15, 35R30     

Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations

We study a discontinuous Galerkin finite element method (DGFEM) for the Stokes equations with a weak stabilization of the viscous term. We prove that, as the stabilization parameter γ tends to infinity, the solution converges at speed γ^?1 to the solution of some stable and well‐known nonconforming finite element methods (NCFEM) for the Stokes equations. In addition, we show that an a posteriori error estimator for the DGFEM‐solution based on the reconstruction of a locally conservative H(div, Ω)‐tensor tends at the same speed to a classical a posteriori error estimator for the NCFEM‐solution. These results can be used to affirm the robustness of the DGFEM‐method and also underline the close relationship between the two approaches. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L²(H¹) and L²(L²) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition k_n≥ch², which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.     

Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

In the spectral Petrov‐Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained. In this paper, we extend this idea to the nonorthogonal dual Bernstein polynomials. A compact general formula is derived for the modal basis functions based on dual Bernstein polynomials. Then, we present a Bernstein‐spectral Petrov‐Galerkin method for a class of time fractional partial differential equations with Caputo derivative. It is shown that the method leads to banded sparse linear systems for problems with constant coefficients. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.