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In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on…
In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new…
We present an efficient numerical method, inspired by transformation optics, for solving the Poisson equation in complex and arbitrarily shaped geometries. The approach operates by mapping the physical domain to a uniform computational…
A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is recalled in this work. The strategy extends the a posteriori error estimation framework introduced by Guignard and…
The solution of the elliptic partial differential equation has interface singularity at the points which are either the intersections of interfaces or the intersections of interfaces with the boundary of the domain. The singularities that…
We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on solvers…
In this paper we study the mixed virtual element approximation to an elliptic optimal control problem with boundary observations. The objective functional of this type of optimal control problem contains the outward normal derivatives of…
Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system…
We propose a high-order finite element method for linear fourth-order elliptic problems that is both nodally bound-preserving and mass-conservative, based on a variational inequality formulation. The method admits an equivalent strictly…
In this paper, a full (nested) multigrid scheme is proposed to solve eigenvalue problems. The idea here is to use the multilevel correction method to transform the solution of eigenvalue problem to a series of solutions of the corresponding…
We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that…
In this paper we use the GeneralizedMultiscale Finite ElementMethod (GMsFEM) framework, introduced in [20], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing…
This paper introduces a nonconforming virtual element method for general second-order elliptic problems with variable coefficients on domains with curved boundaries and curved internal interfaces. We prove arbitrary order optimal…
We consider a mixed dimensional elliptic partial differential equation posed in a bulk domain with a large number of embedded interfaces. In particular, we study well-posedness of the problem and regularity of the solution. We also propose…
In this paper, we propose a domain decomposition method for multiscale second order elliptic partial differential equations with highly varying coefficients. The method is based on a discontinuous Galerkin formulation. We present both a…
We propose the first optimal geometric multigrid solver for hybrid high-order discretizations that can handle arbitrary polytopal agglomeration hierarchies in both two and three dimensions. The key ingredient is the use of modified skeleton…
The multigrid algorithm is a multilevel approach to accelerate the numerical solution of discretized differential equations in physical problems involving long-range interactions. Multiresolution analysis of wavelet theory provides an…
We consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems. Our main result is that these linear systems, which are symmetric and positive semidefinite, are well approximated by…
In this paper we describe in detail the computational algorithm used by our parallel multigrid elliptic equation solver with adaptive mesh refinement. Our code uses truncation error estimates to adaptively refine the grid as part of the…
We present a novel approach for the construction of basis functions to be employed in selective or adaptive h-refined finite element applications with arbitrary-level hanging node configurations. Our analysis is not restricted to…