Related papers: A direct solver with reutilization of previously-c…
In this work we formally derive and prove the correctness of the algorithms and data structures in a parallel, distributed-memory, generic finite element framework that supports h-adaptivity on computational domains represented as…
We present a fast sparse matrix permutation algorithm tailored to linear systems arising from triangle meshes. Our approach produces nested-dissection-style permutations while significantly reducing permutation runtime overhead. Rather than…
We present a new direct logarithmically optimal in theory and fast in practice algorithm to implement the high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson kind. The key points…
We consider a second-order elliptic boundary value problem with strongly monotone and Lipschitz-continuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the…
In this article, we introduce a fast and memory efficient solver for sparse matrices arising from the finite element discretization of elliptic partial differential equations (PDEs). We use a fast direct (but approximate) multifrontal…
Linear solvers are major computational bottlenecks in a wide range of decision support and optimization computations. The challenges become even more pronounced on heterogeneous hardware, where traditional sparse numerical linear algebra…
When solving partial differential equations (PDEs) using finite difference or finite element methods, efficient solvers are required for handling large sparse linear systems. In this paper, a recursive sparse LU decomposition for matrices…
Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the…
In this paper, we propose a novel recovery based finite element method for the Cahn-Hilliard equation. One distinguishing feature of the method is that we discretize the fourth-order differential operator in a standard $C^0$ linear finite…
We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or…
This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF-DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL…
When optimizing a nonlinear objective, one can employ a neural network as a surrogate for the nonlinear function. However, the resulting optimization model can be time-consuming to solve globally with exact methods. As a result, local…
In this paper, a piecewise quadratic finite element method on rectangular grids for the $H^1$ problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of…
The dense matrix resulting from an integral equation (IE) based solution of Maxwell's equations can be compactly represented by an ${\cal H}^2$-matrix. Given a general dense ${\cal H}^2$-matrix, prevailing fast direct solutions involve…
Solving sparse linear systems from discretized PDEs is challenging. Direct solvers have in many cases quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and…
We present the design and implementation details of a geometric multigrid method on adaptively refined meshes for massively parallel computations. The method uses local smoothing on the refined part of the mesh. Partitioning is achieved by…
In this paper we present a finite element method for the direct transcription of constrained non-linear optimal control problems. We prove that our method converges of high order under mild assumptions. Our analysis uses a regularized…
This manuscript presents a new extended linear system for integral equation based techniques for solving boundary value problems on locally perturbed geometries. The new extended linear system is similar to a previously presented technique…
I consider techniques for Berger-Oliger adaptive mesh refinement (AMR) when numerically solving partial differential equations with wave-like solutions, using characteristic (double-null) grids. Such AMR algorithms are naturally recursive,…
We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper. We propose an iterative two-grid algorithm, in which a nonlinear problem is…