Related papers: hyper.deal: An efficient, matrix-free finite-eleme…
Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of…
Phase-field fracture models lead to variational problems that can be written as a coupled variational equality and inequality system. Numerically, such problems can be treated with Galerkin finite elements and primal-dual active set…
Matrix-free geometric multigrid solvers for elliptic PDEs that have been discretised with Higher-order Discontinuous Galerkin (DG) methods are ideally suited to exploit state-of-the-art computer architectures. Higher polynomial degrees…
We present a scalable and efficient iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of hyperbolic partial differential equations. It is an interplay between domain decomposition methods and HDG…
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…
This work is a user guide to the FEMPAR scientific software library. FEMPAR is an open-source object-oriented framework for the simulation of partial differential equations (PDEs) using finite element methods on distributed-memory…
Deep learning has become a popular tool across many scientific fields, including the study of differential equations, particularly partial differential equations. This work introduces the basic principles of deep learning and the Deep…
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 space-time multigrid method based on tensor-product space-time finite element discretizations. The method is facilitated by the matrix-free capabilities of the {\ttfamily deal.II} library. It addresses both high-order…
The simulation of high-dimensional problems with manageable computational resource represents a long-standing challenge. In a series of our recent work [25, 17, 18, 24], a class of sparse grid DG methods has been formulated for solving…
We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical…
This paper introduces JAX-FEM, an open-source differentiable finite element method (FEM) library. Constructed on top of Google JAX, a rising machine learning library focusing on high-performance numerical computing, JAX-FEM is implemented…
Deep learning-based methods have shown remarkable effectiveness in solving PDEs, largely due to their ability to enable fast simulations once trained. However, despite the availability of high-performance computing infrastructure, many…
Numerical solution of partial differential equations on parallel computers using domain decomposition usually requires synchronization and communication among the processors. These operations often have a significant overhead in terms of…
Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering. To achieve optimal performance, solvers have to…
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…
High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial…
We present scalable iterative solvers and preconditioning strategies for Hybridizable Discontinuous Galerkin (HDG) discretizations of partial differential equations (PDEs) on graphics processing units (GPUs). The HDG method is implemented…
Unfitted finite element methods, like CutFEM, have traditionally been implemented in a matrix-based fashion, where a sparse matrix is assembled and later applied to vectors while solving the resulting linear system. With the goal of…
Partial differential equations (PDEs) are often computationally challenging to solve, and in many settings many related PDEs must be be solved either at every timestep or for a variety of candidate boundary conditions, parameters, or…