Adaptive finite element methods for parabolic problems i. This has been out of print for several years, and i have felt a need and been encouraged by colleagues and friends to publish an updated version. A stable spacetime finite element method for parabolic. Weak galerkin mixed finite element methods for parabolic. Both continuous and discontinuous time weak galerkin finite element. The purpose of this thesis is to present some results on. Galerkin method weighted residual methods a weighted residual method uses a finite number of functions.
L convergence of finite element galerkin approximations for parabolic problems by joachim a. Mixed finite element methods on nonmatching multiblock. The main objective of this thesis is to analyze mortar nite element methods for elliptic and parabolic initialboundary value problems. The approach is based on first discretizing in the spatial variables by galerkin s method, using piecewise polynomial trial functions, and then applying some single step or multistep time stepping method. Threelevel galerkin methods for parabolic equations. Superconvergence property of finite element methods for parabolic optimal control problems. Weak galerkin finite element methods for parabolic equations. Weak galerkin finite element method with secondorder. Galerkin finite element methods for parabolic problems math. Pdf weak galerkin finite element methods for parabolic interface. A newly developed weak galerkin method is proposed to solve parabolic. Hou abstract in this paper, we develop a time and its corresponding spatial discretization scheme, based upon the assumption of a certain weak singularity of iiuttlllzn llut112, for the dis continuous galerkin finite element method for.
The differential equation of the problem is du0 on the boundary bu, for. Discontinuous galerkin finite element method for parabolic problems. The approximate solution of parabolic initial boundary value problems. Furthermore, a petrovgalerkin method may be required in the nonsymmetric case.
Superconvergence property of finite element methods for. Dg finite element methods in which time and space variables are adjusted using a posteriori. The lumped mass finite element method for a parabolic problem volume 26 issue 3 c. Galerkin finite element method for parabolic problems. The approach is based on first discretizing in the spatial variables by. Strong superconvergence of finite element methods for. If this is the first time you use this feature, you will be asked to. Finite element methods for parabolic equations semantic scholar. Pdf weak galerkin finite element methods for parabolic equations. Discontinuous galerkin finite element methods for second. The initialboundary value problem for a linear parabolic equation with the.
Typical semidiscrete and fully discrete schemes are. Weak galerkin finite element methods for elliptic and. Weak galerkin finite element methods for elliptic and parabolic problems on polygonal meshes mwndea 2020 naresh kumar department of mathematics indian institute of technology. The extended finite element method xfem is a numerical technique based on the generalized finite element method gfem and the partition of unity method pum. Finite element methods for parabolic problemssome steps in the evolution. A newly developed weak galerkin method is proposed to solve parabolic equations. Journal of industrial and management optimization, vol. The analysis of these methods proceeds in two steps.
We present partially penalized immersed finite element methods for solving parabolic interface problems on cartesian meshes. Galerkin methods have been presented and analyzed for linear and nonlinear parabolic initial boundary value problems 7. For the discretization of a quadratic convex optimal control problem, the state and co. There are many numerical methods available for solving this kind of parabolic problems, including finite element methods 1,2, discontinuous galerkin finite element methods 3,4. Single step methods and rational approximations of semigroups single step fully discrete schemes for the inhomogeneous equation. Partially penalized immersed finite element methods for. The approach is based on first discretizing in the spatial variables by galerkins method, using piecewise polynomial trial functions, and then applying some single step.
Weak galerkin finite element methods for parabolic. Results of numerical experiments will show that without an appropriate modification the standard dg galerkin finite element method applied to a parabolic problem with an inhomogeneous constraint. In this paper, we consider the galerkin finite element method for solving the fractional stochastic diffusionwave equations driven by multiplicative noise, which can be used to describe the. Fulldiscrete weak galerkin finite element method for solving diffusionconvection problem. Galerkin methods for parabolic equations siam journal on. The lumped mass finite element method for a parabolic. In this paper, an adaptive algorithm is presented and analyzed for choosing the. H galerkin mixed finite element methods for elliptic. Finite element approximation of initial boundary value problems. In chapter 2 of this dissertation, we have discussed. An introduction to the finite element method fem for di. Galerkin approximations and finite element methods ricardo g. In this article, interior penalty discontinuous galerkin methods using immersed finite element functions are employed to solve parabolic interface problems.
Abstract pdf 909 kb 1988 finite element methods for parabolic and hyperbolic partial integrodifferential equations. Superconvergence of h1galerkin mixed finite element methods for elliptic optimal control problems chunmei liu1, tianliang hou2. Since the formulation and analysis of galerkin finite element methods for parabolic problems are generally based on ideas and results from the corresponding. Weak galerkin mixed finite element methods for parabolic equations with memory xiaomeng li, qiang xu, and ailing zhu, school of mathematical and statistics, shandong normal university. This book provides insight in the mathematics of galerkin finite element method as applied to parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the. The basis of this work is my earlier text entitled galerkin finite element methods for parabolic problems, springer lecture notes in mathematics, no. The extra cost of using independent approxi mations for both u and i is. In this paper, a finite element method for a parabolic optimal control problem is introduced and analyzed. A mortar finite element space is introduced on the nonmatching interfaces. Typical semidiscrete and fully discrete schemes are presented and analyzed. Incomplete iterative solution of the algebraic systems at the time levels the discontinuous galerkin time stepping method a nonlinear problem. Discontinuous galerkin finite element method for parabolic. Optimal convergence for both semidiscrete and fully discrete schemes is proved.
Download book online more book more links galerkin finite element methods for parabolic problems springer series in computational mathematics download book online more book. The numerical analysis of boundary value problems for partial differential. Discontinuous galerkin immersed finite element methods for parabolic interface problems qing yangyand xu zhangz abstract in this article, interior penalty discontinuous galerkin methods. Math 6630 is the one semester of the graduatelevel introductory course on the numerical methods for partial differential equations pdes. There are many numerical methods available for solving this kind of parabolic problems, including finite element methods, discontinuous galerkin finite element methods, nonconforming. An introduction to the finite element method fem for. Discontinuous galerkin immersed finite element methods for. Galerkin finite element methods for parabolic problems, vol. Bramble and thome 2 consider galerkin methods with parameters h and k tied. This paper is the first part in a series of papers on adaptive finite element methods for parabolic problems. First, we will show that the galerkin equation is a well. Also in the 1970s, but independently, galerkin methods for elliptic and. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids.
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