Two-scale composite finite element method for parabolic problems with smooth and nonsmooth initial data

Pramanick, Tamal; Sinha, Rajen Kumar

doi:10.1007/s12190-017-1153-9

Cited by 4 publications

(6 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now, we define some geometric constants (cf. ) which will be used later in the error analysis. Let x , y ∈ G ( τ ) be the pair of vertices for τ ∈ sons( K ),

K \in {scriptT}_{Γ}

.…”

Section: Cfe Approximations and Auxiliary Resultsmentioning

confidence: 99%

“…In order to discretized the domain Ω, we introduce two‐scale grid: the coarse‐scale and the fine‐scale grid (see, e.g., Refs. [ and ]). The CFE discretization describes in the following steps:…”

Section: Notations and Preliminariesmentioning

confidence: 99%

“…Recently, error estimates for the linear parabolic problem by a two‐scale CFE method is described in Ref. .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

Pramanick

Sinha

2018

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

We study a new class of finite elements so‐called composite finite elements (CFEs), introduced earlier by Hackbusch and Sauter, Numer. Math., 1997; 75:447‐472, for the approximation of nonlinear parabolic equation in a nonconvex polygonal domain. A two‐scale CFE discretization is used for the space discretizations, where the coarse‐scale grid discretized the domain at an appropriate distance from the boundary and the fine‐scale grid is used to resolve the boundary. A continuous, piecewise linear CFE space is employed for the spatially semidiscrete finite element approximation and the temporal discretizations is based on modified linearized backward Euler scheme. We derive almost optimal‐order convergence in space and optimal order in time for the CFE method in the L∞(L2) norm. Numerical experiment is carried out for an L‐shaped domain to illustrate our theoretical findings.

show abstract

“…Now, we define some geometric constants (cf. ) which will be used later in the error analysis. Let x , y ∈ G ( τ ) be the pair of vertices for τ ∈ sons( K ),

K \in {scriptT}_{Γ}

.…”

Section: Cfe Approximations and Auxiliary Resultsmentioning

confidence: 99%

Section: Notations and Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

Pramanick

Sinha

2018

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

show abstract

“…Later the convergence properties of CFEs in the framework of a priori analysis are done. Over the past few years there has been significant research on the CFE method for parabolic problems, as evidenced by the studies conducted, see [22,23,24]. The authors have conducted a study on the error analysis of the CFE method for linear parabolic type problems and presented numerical examples in support of the theoretical error analysis.…”

Section: Literature Reviewmentioning

confidence: 99%

“…Theorem 11. Let U n be the solution of Equation (22) and u be the solution of Equation (1). Let the condition (2) holds true.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Adaptation of the composite finite element framework for semilinear parabolic problems

Anand,

Pramanick

2024

J. Numer. Anal. Approx. Theory

Self Cite

View full text Add to dashboard Cite

In this article, we discuss one of the subsections of finite element method (FEM), classified as the Composite Finite Element Method, abbreviated as CFE. Dimensionality reduction is the primary benefit of the CFE method as it helps to reduce the complexity for the domain space. The degrees of freedom is more in FEM, while compared to the CFE method. Here, the semilinear parabolic problem in a 2D convex polygonal domain is considered. The analysis of the semidiscrete method for the problem is carried out initially in the CFE framework. Here, the discretization would be carried out for the space co-ordinate. Then, fully discrete problem is taken into account, where both the spatial and time components get discretized. In the fully discrete case, the backward Euler method and the Crank-Nicolson method in the CFE framework is adapted for the semilinear problem. The properties of convergence are derived and the error estimates are examined. It is verified that the order of convergence is preserved. The results obtained from the numerical computations are also provided.

show abstract

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

Pramanick

Sinha

2019

Computational Methods in Applied Mathematics

View full text Add to dashboard Cite

The purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the {L^{\infty}(H^{1})}-norm and {L^{\infty}(L^{2})}-norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.

show abstract

Two-scale composite finite element method for parabolic problems with smooth and nonsmooth initial data

Cited by 4 publications

References 10 publications

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

Adaptation of the composite finite element framework for semilinear parabolic problems

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

Contact Info

Product

Resources

About