Related papers: High order two-grid finite difference methods for …
Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes.…
In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality…
In this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified N\'ed\'elec-Raviart-Thomas finite element methods for model problems in $\mathbf{H}(\operatorname{\mathbf{curl}})$ and…
In this paper, a meshfree method using the deep neural network (DNN) approach is developed for solving two kinds of dynamic two-phase interface problems governed by different dynamic partial differential equations on either side of the…
We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers…
In this paper, we study the stability and convergence of a decoupled and linearized mixed finite element method (FEM) for incompressible miscible displacement in a porous media whose permeability and porosity are discontinuous across some…
Multi-level numerical methods that obtain the exact solution of a linear system are presented. The methods are devised by combining ideas from the full multi-grid algorithm and perfect reconstruction filters. The problem is stated as…
In this paper we present an immersed weak Galerkin method for solving second-order elliptic interface problems on polygonal meshes, where the meshes do not need to be aligned with the interface. The discrete space consists of constants on…
A numerical approach for solving evolutionary partial differential equations in two and three space dimensions on block-based adaptive grids is presented. The numerical discretization is based on high-order, central finite-differences and…
In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the…
Harmonic average method has been widely utilized to deal with heterogeneous coefficients in solving differential equations. One remarkable advantage of the harmonic averaging method is that no derivative of the coefficient is needed.…
This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the…
Level set-based immersed boundary techniques operate on nonconforming meshes while providing a crisp definition of interface and external boundaries. In such techniques, an isocontour of a level set field interpolated from nodal level set…
Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted…
This article presents an immersed virtual element method for solving a class of interface problems that combines the advantages of both body-fitted mesh methods and unfitted mesh methods. A background body-fitted mesh is generated…
The present work illustrates a difficulty with the level-set method to accurately capture the curvature of interfaces in regions that are of equal distance to two or more interfaces. Such regions are characterized by kinks in the level-set…
An efficient $hp$-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We…
This paper presents a high-order method for solving an interface problem for the Poisson equation on embedded meshes through a coupled finite element and integral equation approach. The method is capable of handling homogeneous or…
New implicit and implicit-explicit time-stepping methods for the wave equation in second-order form are described with application to two and three-dimensional problems discretized on overset grids. The implicit schemes are single step,…
In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization.…