13,406 research outputs found

    Adaptive FE-BE coupling for strongly nonlinear transmission problems with friction II

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    This article discusses the well-posedness and error analysis of the coupling of finite and boundary elements for transmission or contact problems in nonlinear elasticity. It concerns W^{1,p}-monotone Hencky materials with an unbounded stress-strain relation, as they arise in the modelling of ice sheets, non-Newtonian fluids or porous media. For 1<p<2 the bilinear form of the boundary element method fails to be continuous in natural function spaces associated to the nonlinear operator. We propose a functional analytic framework for the numerical analysis and obtain a priori and a posteriori error estimates for Galerkin approximations to the resulting boundary/domain variational inequality. The a posteriori estimate complements recent estimates obtained for mixed finite element formulations of friction problems in linear elasticity.Comment: 20 pages, corrected typos and improved expositio

    A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems

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    For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using Pk−Pk−1 \mathbb{P}_{k}-\mathbb{P}_{k-1} element (k≥1) (k\geq 1) is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order O(dof−2kd) O(dof^{\frac{-2k}{d}})

    A mixed finite element method for the generalized Stokes problem

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    [Abstract] We present and analyse a new mixed finite element method for the generalized Stokes problem. The approach, which is a natural extension of a previous procedure applied to quasi-Newtonian Stokes flows, is based on the introduction of the flux and the tensor gradient of the velocity as further unknowns. This yields a two-fold saddle point operator equation as the resulting variational formulation. Then, applying a slight generalization of the well known Babuška–Brezzi theory, we prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. In particular, the finite element subspaces providing stability coincide with those employed for the usual Stokes flows except for one of them that needs to be suitably enriched. We also develop an a posteriori error estimate (based on local problems) and propose the associated adaptive algorithm to compute the finite element solutions. Several numerical results illustrate the performance of the method and its capability to localize boundary layers, inner layers, and singularities

    Finite element methods for time-harmonic wave equations

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    This thesis concerns the numerical simulation of time-harmonic wave equations using the finite element method. The main difficulties in solving wave equations are the large number of unknowns and the solution of the resulting linear system. The focus of the research is in preconditioned iterative methods for solving the linear system and in the validation of the result with a posteriori error estimation. Two different solution strategies for solving the Helmholtz equation, a domain decomposition method and a preconditioned GMRES method are studied. In addition, an a posterior error estimate for the Maxwell's equations is presented. The presented domain decomposition method is based on the hybridized mixed Helmholtz equation and using a high-order, tensorial eigenbasis. The efficiency of this method is demonstrated by numerical examples. As the first step towards the mathematical analysis of the domain decomposition method, preconditioners for mixed systems are studied. This leads to a new preconditioner for the mixed Poisson problem, which allows any preconditioned for the first order finite element discretization of the Poisson problem to be used with iterative methods for the Schur complement problem. Solving the linear systems arising from the first order finite element discretization of the Helmholtz equation using the GMRES method with a Laplace, an inexact Laplace, or a two-level preconditioner is discussed. The convergence properties of the preconditioned GMRES method are analyzed by using a convergence criterion based on the field of values. A functional type a posterior error estimate is derived for simplifications of the Maxwell's equations. This estimate gives computable, guaranteed upper bounds for the discretization error

    An Adaptive Mixed Finite Element Method using the Lagrange Multiplier Technique

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    Adaptive methods in finite element analysis are essential tools in the efficient computation and error control of problems that may exhibit singularities. In this paper, we consider solving a boundary value problem which exhibits a singularity at the origin due to both the structure of the domain and the regularity of the exact solution. We introduce a hybrid mixed finite element method using Lagrange Multipliers to initially solve the partial differential equation for the both the flux and displacement. An a posteriori error estimate is then applied both locally and globally to approximate the error in the computed flux with that of the exact flux. Local estimation is the key tool in identifying where the mesh should be refined so that the error in the computed flux is controlled while maintaining efficiency in computation. Finally, we introduce a simple refinement process in order to improve the accuracy in the computed solutions. Numerical experiments are conducted to support the advantages of mesh refinement over a fixed uniform mesh
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