Stochastic Galerkin methods for elliptic interface problems with random input

In this work, we consider random elliptic interface problems, namely, the media in elliptic equations have both randomness and interfaces. A Galerkin method using bi-orthogonal polynomials is used to convert the random problem into an uncoupled system of deterministic interface problems. A principle on how to choose the orders of the approximated polynomial spaces is given based on the sensitivity analysis in random spaces, with which the total degree of freedom can be significantly reduced. Then immersed finite element methods are introduced to solve the resulting system. Convergence results are given both theoretically and numerically.     

An analysis of discontinuous Galerkin methods for elliptic problems

We study global and local behaviors for three kinds of discontinuous Galerkin schemes for elliptic equations of second order. We particularly investigate several a posteriori error estimations for the discontinuous Galerkin schemes. These theoretical results are applied to develop local/parallel and adaptive finite element methods, based on the discontinuous Galerkin methods. Dedicated to Dr. Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem Mathematics subject classifications (2000) 65N12, 65N15, 65N30. Aihui Zhou: Subsidized by the Special Funds for Major State Basic Research Projects, and also partially supported by National Science Foundation of China. Reinhold Schneider: Supported in part by DFG Sonderforschungsbereich SFB 393. Yuesheng Xu: Correspondence author. Supported in part by the US National Science Foundation under grants DMS-9973427 and CCR-0312113, by Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under program “Hundreds Distinguished Young Chinese Scientists”.     

BDDC methods for discontinuous Galerkin discretization of elliptic problems

A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using special partitions of unity and edge constraints. Under certain assumptions on the coefficients and the mesh sizes across ∂Ω_i, where the Ω_i are disjoint subregions of the original region Ω, a condition number estimate C(1+max_ilog(H_i/h_i))² is established with C independent of h_i, H_i and the jumps of the coefficients. The algorithms are well suited for parallel computations and can be straightforwardly extended to the 3-D problems. Results of numerical tests are included which confirm the theoretical results and the necessity of the imposed assumptions.     

Abstract theory of hybridizable discontinuous Galerkin methods for second-order quasilinear elliptic problems

An abstract theory for discretizations of second-order quasilinear elliptic problems based on the mixed-hybrid discontinuous Galerkin method. Discrete schemes are formulated in terms of approximations of the solution to the problem, its gradient, flux, and the trace of the solution on the interelement boundaries. Stability and optimal error estimates are obtained under minimal assumptions on the approximating space. It is shown that the schemes admit an efficient numerical implementation.     

An extended Galerkin analysis for elliptic problems

A general analysis framework is presented in this paper for many different types of finite element methods(including various discontinuous Galerkin methods). For the second-order elliptic equation-div(α▽u) = f, this framework employs four different discretization variables, uh, p_h, uhand p_h, where uh and phare for approximation of u and p =-α▽u inside each element, and uhand phare for approximation of the residual of u and p·n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.     

Stochastic Galerkin techniques for random ordinary differential equations

Over the last decade the stochastic Galerkin method has become an established method to solve differential equations involving uncertain parameters. It is based on the generalized Wiener expansion of square integrable random variables. Although there exist very sophisticated variants of the stochastic Galerkin method (wavelet basis, multi-element approach) convergence for random ordinary differential equations has rarely been considered analytically. In this work we develop an asymptotic upper boundary for the L ₂-error of the stochastic Galerkin method. Furthermore, we prove convergence of a local application of the stochastic Galerkin method and confirm convergence of the multi-element approach within this context.     

Two-grid methods of finite element solutions for semi-linear elliptic interface problems

Numerical Algorithms - In this paper, we present two efficient two-grid algorithms for solving two-dimensional semi-linear elliptic interface problems using finite element method. To linearize the...     

Jacobi spectral Galerkin method for elliptic Neumann problems

This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489–1505, 1994) and Auteri et al. (J Comput Phys 185:427–444, 2003), based on Legendre polynomials, to Jacobi polynomials with arbitrary α and β. The key to the efficiency of our algorithms is to construct appropriate basis functions with zero slope at the endpoints, which lead to systems with sparse matrices for the discrete variational formulations. The direct solution algorithm developed for the homogeneous Neumann problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.     

Finite element methods and their convergence for elliptic and parabolic interface problems

In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in two-dimensional convex polygonal domains. Nearly the same optimal -norm and energy-norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical. Received July 7, 1996 / Revised version received March 3, 1997     

Error estimates for a discontinuous Galerkin method for elliptic problems

In this paper, we consider an elliptic problem with the homogeneous Dirichlet boundary condition and introduce discontinuous Galerkin approximations of the problem. Optimal error estimates of discontinuous Galerkin approximations are obtained.     

Local discontinuous Galerkin method for parabolic interface problems

In this paper, the minimal dissipation local discontinuous Galerkin method is studied to solve the parabolic interface problems in two-dimensional convex polygonal domains. The interface may be arbitrary smooth curves. The proposed method is proved to be L2 stable and the order of error estimates in the given norm is O(h|logh1|/2). Numerical experiments show the efficiency and accuracy of the method.     

Bubble stabilized discontinuous Galerkin method for parabolic and elliptic problems

In this paper we give an analysis of a bubble stabilized discontinuous Galerkin method for elliptic and parabolic problems. The method consists of stabilizing the numerical scheme by enriching the discontinuous affine finite element space elementwise by quadratic bubbles. This approach leads to optimal convergence in the space and time discretization parameters.     

The discrete maximum principle for Galerkin solutions of elliptic problems

This paper provides an equivalent characterization of the discrete maximum principle for Galerkin solutions of general linear elliptic problems. The characterization is formulated in terms of the discrete Green’s function and the elliptic projection of the boundary data. This general concept is applied to the analysis of the discrete maximum principle for the higher-order finite elements in one-dimension and to the lowest-order finite elements on simplices of arbitrary dimension. The paper surveys the state of the art in the field of the discrete maximum principle and provides new generalizations of several results.     

Stochastic methods for Dirichlet problems

Using an equivalent expression for solutions of second order Dirichlet problems in terms of Ito type stochastic differential equations, we develop a numerical solution method for Dirichlet boundary value problems. It is possible with this idea to solve for solution values of a partial differential equation at isolated points without having to construct any kind of mesh and without knowing approximations for the solution at any other points. Our method is similar to a recently published approach, but differs primarily in the handling of the boundary. Some numerical examples are presented, applying these techniques to model Laplace and Poisson equations on the unit disk. Visiting Professor, Universidad de Salamanca.     

Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case

Summary. In this paper we analyze a family of discontinuous Galerkin methods, parameterized by two real parameters, for elliptic problems in one dimension. Our main results are: (1) a complete inf-sup stability analysis characterizing the parameter values yielding a stable scheme and energy norm error estimates as a direct consequence thereof, (2) an analysis of the error in L² where the standard duality argument only works for special parameter values yielding a symmetric bilinear form and different orders of convergence are obtained for odd and even order polynomials in the nonsymmetric case. The analysis is consistent with numerical results and similar behavior is observed in two dimensions.Mathematics Subject Classification (2000): 65M60, 65M15Research supported by: The Swedish Foundation for International Cooperation in Research and Higher Education     

Discontinuous Galerkin isogeometric analysis for elliptic problems with discontinuous diffusion coefficients on surfaces

Numerical Algorithms - This paper is concerned with using discontinuous Galerkin isogeometric analysis (dG-IGA) as a numerical treatment of diffusion problems on orientable surfaces ${Omega }...     

Collocation methods for linear elliptic problems

Collocation methods based on piecewise Hermite cubic polynomials are applied to linear elliptic problems subject to Dirichlet and Neumann boundary conditions on rectangular domains. A priori estimates are obtained for the error of approximation.