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The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
We consider the numerical solution of Poisson's equation on structured grids using geometric multigrid with nonstandard coarse grids and coarse level operators. We are motivated by the problem of developing high-order accurate numerical…
Floorplanning is the first stage of VLSI physical design. An effective floorplanning engine definitely has positive impact on chip design speed, quality and performance. In this paper, we present a novel mathematical model to characterize…
In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method…
We show that a generalised sparse grid combination technique which combines multi-variate extrapolation of finite difference solutions with the standard combination formula lifts a second order accurate scheme on regular meshes to a fourth…
A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the…
We introduce a finite element method for numerical upscaling of second order elliptic equations with highly heterogeneous coefficients. The method is based on a mixed formulation of the problem and the concepts of the domain decomposition…
We give sufficient conditions under which the convergence of finite difference approximations in the space variable of possibly degenerate second order parabolic and elliptic equations can be accelerated to any given order of convergence by…
In this paper, a novel parallel hybrid iterative method is proposed for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of variational inequalities for inverse strongly monotone…
We develop a method to compute the $H^2$-conforming finite element approximation to planar fourth order elliptic problems without having to implement $C^1$ elements. The algorithm consists of replacing the original $H^2$-conforming scheme…
In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a…
We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial…
In this paper, a parameter-uniform fitted mesh finite difference scheme is constructed and analyzed for a class of singularly perturbed interior turning point problems. The solution of this class of turning point problem possess two outflow…
A homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such…
This paper aims to study the convergence of adaptive finite element method for control constrained elliptic optimal control problems under $L^2$-norm. We prove the contraction property and quasi-optimal complexity for the $L^2$-norm errors…
In this paper, a two-grid method is proposed to linearize and symmetrize the steady-state Poisson-Nernst-Planck equations. The computational system is decoupled to linearize and symmetrize equations by using this method, which can improve…
We propose a multiscale approach for an elliptic multiscale setting with general unstructured diffusion coefficients that is able to achieve high-order convergence rates with respect to the mesh parameter and the polynomial degree. The…
We investigate multiscale finite element methods for an elliptic distributed optimal control problem with rough coefficients. They are based on the (local) orthogonal decomposition methodology of M\aa lqvist and Peterseim.
The use of multigrid and related preconditioners with the finite element method is often limited by the difficulty of applying the algorithm effectively to a problem, especially when the domain has a complex shape or adaptive refinement. We…
This paper describes a massively parallel algebraic multigrid method based on non-smoothed aggregation. It is especially suited for solving heterogeneous elliptic problems as it uses a greedy heuristic algorithm for the aggregation that…