Related papers: Geometric scaling: a simple preconditioner for cer…
In the context of isogeometric analysis, we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self-adjoint. These are the collocation method and the…
This Paper conducts a thorough simulation study to assess the effectiveness of various acceleration techniques designed to enhance the conjugate gradient algorithm, which is used for solving large linear systems to accelerate Bayesian…
Unfitted finite element methods, e.g., extended finite element techniques or the so-called finite cell method, have a great potential for large scale simulations, since they avoid the generation of body-fitted meshes and the use of graph…
Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions. The one-level domain decomposition preconditioners are based on the…
In [McDonald, Pestana and Wathen, \textit{SIAM J. Sci. Comput.}, 40 (2018), pp. A1012--A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial differential equations, in which…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned…
This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework…
Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered.…
This paper introduces a geometric multigrid preconditioner for the Shifted Boundary Method (SBM) designed to solve PDEs on complex geometries. While SBM simplifies mesh generation by using a non-conforming background grid, it often results…
The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite difference like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using…
Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…
Solutions to many partial differential equations satisfy certain bounds or constraints. For example, the density and pressure are positive for equations of fluid dynamics, and in the relativistic case the fluid velocity is upper bounded by…
We present a new line-based discontinuous Galerkin (DG) discretization scheme for first- and second-order systems of partial differential equations. The scheme is based on fully unstructured meshes of quadrilateral or hexahedral elements,…
We present a novel linear solver that works well for large systems obtained from discretizing PDEs. It is robust and, for the examples we studied, the computational effort scales linearly with the number of equations. The algorithm is based…
Two-level domain decomposition (DD) methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level…
We present a scalable approach to solve a class of elliptic partial differential equation (PDE)-constrained optimization problems with bound constraints. This approach utilizes a robust full-space interior-point (IP)-Gauss-Newton…
A modification of the generalized shift-splitting (GSS) method is presented for solving singular saddle point problems. In this kind of modification, the diagonal shift matrix is replaced by a block diagonal matrix which is symmetric…
The construction of robust solvers for linear systems obtained from the discretization of partial differential equations using Isogeometric Analysis is challenging since the condition number of the system matrix not only grows with the…
A method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely…