Related papers: A Fourth-Order Cut-cell Multigrid Method for Solvi…
We propose a fast fourth-order cut cell method for solving constant-coefficient elliptic equations in two-dimensional irregular domains. In our methodology, the key to dealing with irregular domains is the poised lattice generation (PLG)…
We propose a fourth-order cut-cell method for solving Poisson's equations in three-dimensional irregular domains. Major distinguishing features of our method include (a) applicable to arbitrarily complex geometries, (b) high order…
We present multigrid methods for solving elliptic partial differential equations on arbitrary domains using the nodal ghost finite element method, an unfitted boundary approach where the domain is implicitly defined by a level-set function.…
A parallel cut-cell algorithm is described to solve the free-boundary problem of the Grad-Shafranov equation. The algorithm reformulates the free-boundary problem in an irregular bounded domain and its important aspects include a searching…
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 propose a new numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds. One advantage of this method is its ability to bypass the need for global triangulations or grids on the manifolds.…
We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…
In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is…
As part of our development of a computer code to perform 3D `constrained evolution' of Einstein's equations in 3+1 form, we discuss issues regarding the efficient solution of elliptic equations on domains containing holes (i.e., excised…
It is well known that multigrid methods are optimally efficient for solution of elliptic equations (O(N)), which means that effort is proportional to the number of points at which the solution is evaluated). Thus this is an ideal method to…
A method is presented to include irregular domain boundaries in a geometric multigrid solver. Dirichlet boundary conditions can be imposed on an irregular boundary defined by a level set function. Our implementation employs quadtree/octree…
We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving optimization problem on the finest finite element space is…
A new numerical domain decomposition method is proposed for solving elliptic equations on compact Riemannian manifolds. The advantage of this method is to avoid global triangulations or grids on manifolds. Our method is numerically tested…
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…
Unfitted finite element methods have emerged as a popular alternative to classical finite element methods for the solution of partial differential equations and allow modeling arbitrary geometries without the need for a boundary-conforming…
We propose a fourth-order cut-cell method for solving the two-dimensional advection-diffusion equation with moving boundaries on a Cartesian grid. We employ the ARMS technique to give an explicit and accurate representation of moving…
In this work, we develop algebraic solvers for linear systems arising from the discretization of second-order elliptic partial differential equations by saddle-point mixed finite element methods of arbitrary polynomial degree $p \ge 0$ on…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE…
Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective…