Related papers: A Multilevel Correction Type of Adaptive Finite El…
A Novel Scaled boundary finite element method, initially developed in Civil Engineering, is reformulated for solving boundary value problems in computational electromagnetics.
The paper presents a finite element scheme for the elastic transmission eigenvalue problem written as a fourth order eigenvalue problem. The scheme uses piecewise cubic polynomials and obtains optimal convergence rate. Compared with other…
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…
We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on…
Proofs of convergence of adaptive finite element methods for the approximation of eigenvalues and eigenfunctions of linear elliptic problems have been given in a several recent papers. A key step in establishing such results for multiple…
This article is devoted to computing the lower and upper bounds of the Laplace eigenvalue problem. By using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extension $Q_1^{\rm rot}$, we get the lower…
Numerical methods for the transmission eigenvalue problems are hot topics in recent years. Based on the work of Lin and Xie [Math. Comp., 84(2015), pp. 71-88], we build a multigrid method to solve the problems. With our method, we only need…
We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
A numerical method is proposed for a class of stochastic control problems including singular behavior. This method solves an infinite-dimensional linear program equivalent to the stochastic control problem using a finite element type…
We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…
The scaled boundary finite element method is known for its capability in reproducing highly-detailed solution fields. This, however, is only attainable in those cases where analytical solutions exist. Many others invoke the use of numerical…
This paper introduces a new computational methodology for determining a-posteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finite-element…
In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…
This paper is concerned with the computable error estimates for the eigenvalue problem which is solved by the general conforming finite element methods on the general meshes. Based on the computable error estimate, we can give an…
We propose a new method to design adaptation algorithms that guarantee a certain prescribed level of performance and are applicable to systems with nonconvex parameterization. The main idea behind the method is, given the desired…
Modeling of physical systems includes extensive use of software packages that implement the accurate finite element method for solving differential equations considered along with the appropriate initial and boundary conditions. When the…
In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several…
This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection…
In this paper, we study the adaptive planewave discretization for a cluster of eigenvalues of second-order elliptic partial differential equations. We first design an a posteriori error estimator and prove both the upper and lower bounds.…