Related papers: High level implementation of geometric multigrid s…
Numerical solution of discrete PDEs corresponding to saddle point problems is highly relevant to physical systems such as Stokes flow. However, scaling up numerical solvers for such systems is often met with challenges in efficiency and…
When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…
Topology optimization for large scale problems continues to be a computational challenge. Several works exist in the literature to address this topic, and all make use of iterative solvers to handle the linear system arising from the Finite…
The numerical simulation of cardiac electrophysiology is a highly challenging problem in scientific computing. The Bidomain system is the most complete mathematical model of cardiac bioelectrical activity. It consists of an elliptic and a…
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…
As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients…
Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system…
Fully realizing the potential of multigrid solvers often requires custom algorithms for a given application model, discretizations and even regimes of interest, despite considerable effort from the applied math community to develop fully…
The automated finite element analysis of complex CAD models using boundary-fitted meshes is rife with difficulties. Immersed finite element methods are intrinsically more robust but usually less accurate. In this work, we introduce an…
In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that…
We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…
The combination of machine learning and physical laws has shown immense potential for solving scientific problems driven by partial differential equations (PDEs) with the promise of fast inference, zero-shot generalisation, and the ability…
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.…
Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have become increasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point and…
We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank…
Many of the most fundamental laws of nature can be formulated as partial differential equations (PDEs). Understanding these equations is, therefore, of exceptional importance for many branches of modern science and engineering. However,…
High level domain specific languages for the finite element method underpin high productivity programming environments for simulations based on partial differential equations (PDE) while employing automatic code generation to achieve high…
We present an efficient, robust and fully GPU-accelerated aggregation-based algebraic multigrid preconditioning technique for the solution of large sparse linear systems. These linear systems arise from the discretization of elliptic PDEs.…
This paper builds on the algebraic theory in the companion paper [Algebraic Error Analysis for Mixed-Precision Multigrid Solvers] to obtain discretization-error-accurate solutions for linear elliptic partial differential equations (PDEs) by…
We construct an algebraic multigrid (AMG) based preconditioner for the reduced Hessian of a linear-quadratic optimization problem constrained by an elliptic partial differential equation. While the preconditioner generalizes a geometric…