Convergence of an adaptive mixed finite element method for general second order linear elliptic problems

The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by the non-symmetric and indefinite form of the problem along with the lack of the orthogonality property in mixed finite element methods. The important tools in the analysis are a posteriori error estimators, quasi-orthogonality property and quasi-discrete reliability established using representation formula for the lowest-order Raviart-Thomas solution in terms of the Crouzeix-Raviart solution of the problem. An adaptive marking in each step for the local refinement is based on the edge residual and volume residual terms of the a posteriori estimator. Numerical experiments confirm the theoretical analysis.Comment: 24 pages, 8 figure

Global existence of solutions to Keller--Segel chemotaxis system with heterogeneous logistic source and nonlinear secretion

We study the following Keller-Segel chemotaxis system with logistic source and nonlinear secretion. For this system, we prove the global existence of solutions under suitable assumptions

A Nonconforming Finite Element Approximation for Optimal Control of an Obstacle Problem

The article deals with the analysis of a nonconforming finite element method for the discretization of optimization problems governed by variational inequalities. The state and adjoint variables are discretized using Crouzeix-Raviart nonconforming finite elements, and the control is discretized using a variational discretization approach. Error estimates have been established for the state and control variables. The results of numerical experiments are presented

Edge Patch-Wise Local Projection Stabilized Nonconforming FEM for the Oseen Problem

In finite element approximation of the Oseen problem, one needs to handle two major difficulties, namely, the lack of stability due to convection dominance and the incompatibility between the approximating finite element spaces for the velocity and the pressure. These difficulties are addressed in this article by using an edge patch-wise local projection (EPLP) stabilization technique. The article analyses the EPLP stabilized nonconforming finite element methods for the Oseen problem. For approximating the velocity, the lowest-order Crouzeix-Raviart (CR) nonconforming finite element space is considered; whereas for approximating the pressure, two discrete spaces are considered, namely, the piecewise constant polynomial space and the lowest-order CR finite element space. The proposed discrete weak formulation is a combination of the standard Galerkin method, EPLP stabilization and weakly imposed boundary condition by using Nitsche's technique. The resulting bilinear form satisfies an inf-sup condition with respect to EPLP norm, which leads to the well-posedness of the discrete problem. A priori error analysis assures the optimal order of convergence in both the cases, that is, order one in the case of piecewise constant approximation and 3/2 in the case of CR-finite element approximation for pressure. The numerical experiments illustrate the theoretical findings

An Error Analysis of Discontinuous Finite Element Methods for the Optimal Control Problems Governed by Stokes Equation

In this article, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. A priori error estimates of optimal order are derived for velocity and pressure in the energy norm and the L-2-norm, respectively. Moreover, a reliable and efficient a posteriori error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix-Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces. The theoretical results are illustrated by the numerical experiments

Quasi-Optimality of Adaptive Mixed FEMs for Non-selfadjoint Indefinite Second-Order Linear Elliptic Problems

The well-posedness and the a priori and a posteriori error analysis of the lowest-order Raviart-Thomas mixed finite element method (MFEM) has been established for non-selfadjoint indefinite second-order linear elliptic problems recently in an article by Carstensen, Dond, Nataraj and Pani (Numer. Math., 2016). The associated adaptive mesh-refinement strategy faces the difficulty of the flux error control in H(div, Omega) and so involves a data-approximation error parallel to f - Pi(0)f parallel to in the L-2 norm of the right-hand side f and its piecewise constant approximation Pi(0)f. The separate marking strategy has recently been suggested with a split of a Dorfler marking for the remaining error estimator and an optimal data approximation strategy for the appropriate treatment of parallel to f -Pi(0)f parallel to(L)2((Omega)). The resulting strategy presented in this paper utilizes the abstract algorithm and convergence analysis of Carstensen and Rabus (SINUM, 2017) and generalizes it to general second-order elliptic linear PDEs. The argument for the treatment of the piecewise constant displacement approximation u(RT) is its supercloseness to the piecewise constant approximation Pi(0u) of the exact displacement u. The overall convergence analysis then indeed follows the axioms of adaptivity for separate marking. Some results on mixed and nonconforming finite element approximations on the multiply connected polygonal 2D Lipschitz domain are of general interest