On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.MSC:35J30, 35J40.

Scaling function and wavelet preconditioners for second order elliptic problems

(1981). Second order elliptic boundary value problems in a domain with edges. Communications in Partial Differential Equations: Vol. 6, No. 9, pp. 1023-1042.

This paper is concerned with the phenomenon of gradient superconvergence of finite element approximations to the solutions of two-dimensional second order elliptic boundary value problems. The original concept of gradient superconvergence is that there exist certain points in each element at which the gradient of the finite element approximation has a higher rate of convergence to the gradient of the true solution in a discrete analogue of a Lebesgue norm than that found globally in the Lebesgue norm itself.

The Variational Problems and Equivalent Forms for a Second Order Elliptic Boundary Value Problem

We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in R N \mathbb {R}^{N} . In a sense to be discussed below these sharpen known quasi–optimal L ∞ L_{\infty } and W ∞ 1 W^{1}_{\infty } estimates for the error on irregular quasi–uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u u . We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in R N \mathbb {R}^{N} and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non–smooth problems.

The asymptotics of solutions to scalar second order elliptic boundary value problems in three-dimensional polyhedral domains in the vicinity of an edge is provided in an explicit form. It involves a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the edges. Utilizing the explicit structure of the solution in the vicinity of the edge we present a new method for the extraction of the EFIFs called quasidual function method. It can be interpreted as an extension of the dual function contour integral method in 2-D domains, and involves the computation of a surface integral J[R] along a cylindrical surface of radius R away from the edge as presented in a general framework in (Costabel et al., 2004). The surface integral J[R] utilizes special constructed extraction polynomials together with the dual eigen-functions for extracting EFIFs. This accurate and efficient method provides a polynomial approximation of the EFIF along the edge whose order is adaptively increased so to approximate the exact EFIF. It is implemented as a post-solution operation in conjunction with the p-version finite element method. Numerical realization of some of the anticipated properties of the J[R] are provided, and it is used for extracting EFIFs associated with different scalar elliptic equations in 3-D domains, including domains having edge and vertex singularities. The numerical examples demonstrate the efficiency, robustness and high accuracy of the proposed quasi-dual function method, hence its potential extension to elasticity problems.

We consider the interior and Dirichiet problems and problems with first order boundary conditions, for a second order homogeneous elliptic partial differential operator with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. From there we obtain a priori estimates and regularity results for boundary value problems in Sobolev spaces.

Uniqueness for sublinear boundary value problems

We study the question of approximability for the inverse of the FEM stiffness matrix for (scalar) second order elliptic boundary value problems by blockwise low rank matrices such as those given by the H-matrix format. We show that exponential convergence in the local block rank can be achieved. We also show that exponentially accurate LU-decompositions in the H-matrix format are possible for the stiffness matrices arising in the FEM. Unlike prior works, our analysis avoids any coupling of the block rank r and the mesh width h and also covers mixed Dirichlet-Neumann-Robin boundary conditions.

SummaryThe main result in this paper is that the solution operator for the bi-Laplace problem with zero Dirichlet boundary conditions on a bounded smooth 2d-domain can be split in a positive part and a possibly negative part which both satisfy the zero boundary condition. Moreover, the positive part contains the singularity and the negative part inherits the full regularity of the boundary. Such a splitting allows one to find a priori estimates for fourth order problems similar to the one proved via the maximum principle in second order elliptic boundary value problems. The proof depends on a careful approximative fill-up of the domain by a finite collection of limaçons. The limaçons involved are such that the Green function for the Dirichlet bi-Laplacian on each of these domains is strictly positive.

The original additive average Schwarz method and its variants were introduced for solving second order elliptic boundary value problems. In this paper, we extend the original idea to fourth order problems by designing a variant of the additive average Schwarz method for the plate bending problem using the nonconforming Morley finite element for the discretization. Like the original average method this method uses only nonoverlapping subdomains, and its coarse space does not need any explicit coarse mesh. An analysis of the method, and some numerical experiments to validate the theory are presented.

Let u respective uh be the solution of a second order elliptic boundary value problem respective the corresponding Ritz approximations of finite elements Sh. The error eh=u-uh can be splitted: eh=th+Eh. According to the local regularity of the solution Eh has the corresponding good asymptotic convergence whereas the pollution-term th caused by corners of the boundary is an element of a certain linear space T independent of h.

We consider second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of Lagrange multipliers. This is combined with a fictitious domain approach into which the physical domain is embedded. The resulting saddle point problem will be discretized in terms of wavelets, resulting in an operator equation in l2. Stability of the discretization and consequently the uniform boundedness of the condition number of the finite-dimensional operator independent of the discretization is guaranteed by an appropriate LBB condition. For the iterative solution of the saddle point system, an incomplete Uzawa algorithm is employed. It can be shown that the iterative scheme combined with a nested iteration strategy is asymptotically optimal in the sense that it provides the solution up to discretization error on discretization level J in an overall amount of iterations of order O(N J ), where N J is the number of unknowns on level J. Finally, numerical results are provided.