The subject of this work is to propose adaptive finite element methods based on an optimal maximum norm error control estimate.Using estimators of the local regularity of the unknown exact solution derived from comput...The subject of this work is to propose adaptive finite element methods based on an optimal maximum norm error control estimate.Using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions,the proposed procedures are analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that they generate the correct type of refinement and lead to the desired control under consideration.展开更多
Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind sch...Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.展开更多
An adaptive finite element procedure designed for specific computational goals is presented,using mesh refinement strategies based on optimal or nearly optimal a priori error estimates for the finite element method an...An adaptive finite element procedure designed for specific computational goals is presented,using mesh refinement strategies based on optimal or nearly optimal a priori error estimates for the finite element method and using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions.The proposed procedure is analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that the method can generate the correct type of refinements and lead to the desired control under consideration.展开更多
The subject of the work is to propose a series of papers about adaptive finite element methods based on optimal error control estimate. This paper is the third part in a series of papers on adaptive finite element met...The subject of the work is to propose a series of papers about adaptive finite element methods based on optimal error control estimate. This paper is the third part in a series of papers on adaptive finite element methods based on optimal error estimates for linear elliptic problems on the concave corner domains. In the preceding two papers (part 1:Adaptive finite element method based on optimal error estimate for linear elliptic problems on concave corner domain; part 2:Adaptive finite element method based on optimal error estimate for linear elliptic problems on nonconvex polygonal domains), we presented adaptive finite element methods based on the energy norm and the maximum norm. In this paper, an important result is presented and analyzed. The algorithm for error control in the energy norm and maximum norm in part 1 and part 2 in this series of papers is based on this result.展开更多
Homogenization is concerned with obtaining the average properties of a material. The problem on its own has no easy solution, except in cases like the periodic case, when it can be obtained in closed form. In this pap...Homogenization is concerned with obtaining the average properties of a material. The problem on its own has no easy solution, except in cases like the periodic case, when it can be obtained in closed form. In this paper we consider a numerical solution of the elliptic homogenization problem for the case of rapidly varying tensor or boundary conditions. The method makes use of an adaptive finite element method to correctly capture the rapid change in the tensor or boundary condition. In the numerical experiments we vary the mesh size and do a posteriori error analysis on test problems.展开更多
文摘The subject of this work is to propose adaptive finite element methods based on an optimal maximum norm error control estimate.Using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions,the proposed procedures are analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that they generate the correct type of refinement and lead to the desired control under consideration.
基金supported by the Department of Science & Technology, Government of India under research grant SR/S4/MS:318/06.
文摘Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.
文摘An adaptive finite element procedure designed for specific computational goals is presented,using mesh refinement strategies based on optimal or nearly optimal a priori error estimates for the finite element method and using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions.The proposed procedure is analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that the method can generate the correct type of refinements and lead to the desired control under consideration.
文摘The subject of the work is to propose a series of papers about adaptive finite element methods based on optimal error control estimate. This paper is the third part in a series of papers on adaptive finite element methods based on optimal error estimates for linear elliptic problems on the concave corner domains. In the preceding two papers (part 1:Adaptive finite element method based on optimal error estimate for linear elliptic problems on concave corner domain; part 2:Adaptive finite element method based on optimal error estimate for linear elliptic problems on nonconvex polygonal domains), we presented adaptive finite element methods based on the energy norm and the maximum norm. In this paper, an important result is presented and analyzed. The algorithm for error control in the energy norm and maximum norm in part 1 and part 2 in this series of papers is based on this result.
文摘Homogenization is concerned with obtaining the average properties of a material. The problem on its own has no easy solution, except in cases like the periodic case, when it can be obtained in closed form. In this paper we consider a numerical solution of the elliptic homogenization problem for the case of rapidly varying tensor or boundary conditions. The method makes use of an adaptive finite element method to correctly capture the rapid change in the tensor or boundary condition. In the numerical experiments we vary the mesh size and do a posteriori error analysis on test problems.
