Related papers: A high-performance elliptic solver for plasma boun…
We discuss the efficient implementation of a high-performance second-order collocation-type finite-element scheme for solving the compressible Euler equations of gas dynamics on unstructured meshes. The solver is based on the convex…
In this paper, we present an interior penalty discontinuous Galerkin finite element scheme for solving diffusion problems with strong anisotropy arising in magnetized plasmas for fusion applications. We demonstrate the accuracy produced by…
We have developed an adaptive multigrid code for solving the Poisson equation in gravitational simulations. Finer rectangular subgrids are adaptively created in locations where the density exceeds a local level-dependent threshold. We…
In this work, we consider the solution of boundary integral equations by means of a scalable hierarchical matrix approach on clusters equipped with graphics hardware, i.e. graphics processing units (GPUs). To this end, we extend our…
We present a generalisation of the all-Mach solver of Fuster & Popinet (2018) to account for heat diffusion between two different compressible phases. By solving a two-way coupled system of equations for pressure and temperature, the…
In this paper we present a multigrid approach to solve the Poisson equation in arbitrary domain (identified by a level set function) and mixed boundary conditions. The discretization is based on finite difference scheme and ghost-cell…
A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used…
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…
In this work we consider Runge-Kutta discontinuous Galerkin methods (RKDG) for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on…
The elliptic interface problems with discontinuous and high-contrast coefficients appear in many applications and often lead to huge condition numbers of the corresponding linear systems. Thus, it is highly desired to construct high order…
Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the…
With the rapid advancement of graphical processing units, Physics-Informed Neural Networks (PINNs) are emerging as a promising tool for solving partial differential equations (PDEs). However, PINNs are not well suited for solving PDEs with…
Elliptic Partial Differential Equations (PDEs) play a central role in computing the equilibrium conditions of physical problems (heat, gravitation, electrostatics, etc.). Efficient solutions to elliptic PDEs are also relevant to computer…
We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for…
Efficiently solving large-scale linear systems is a critical challenge in electromagnetic simulations, particularly when using the Crank-Nicolson Finite-Difference Time-Domain (CN-FDTD) method. Existing iterative solvers are commonly…
We develop a spectral low-mode reduced solver for second-order elliptic boundary value problems with spatially varying diffusion coefficients. The approach projects standard finite difference or finite element discretization onto a global…
Many problems in geophysical and atmospheric modelling require the fast solution of elliptic partial differential equations (PDEs) in "flat" three dimensional geometries. In particular, an anisotropic elliptic PDE for the pressure…
We present an update of the General-relativistic multigrid numerical (Gmunu) code, a parallelized, multi-dimensional curvilinear, general relativistic magnetohydrodynamics code with an efficient non-linear cell-centred multigrid (CCMG)…
We introduce the first-order system proximal Galerkin (FOSPG) method, a locally mass-conserving, hybridizable finite element method for solving heterogeneous anisotropic diffusion and obstacle problems. Like other proximal Galerkin methods,…
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…