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In this work, we develop a new algorithm to solve large‐scale incompressible time‐dependent fluid–structure interaction problems using a matrix‐free finite element method in arbitrary Lagrangian–Eulerian frame of reference. We derive a semi‐implicit time integration scheme which improves the geometry‐convective explicit scheme for problems involving the interaction between incompressible hyperelastic solids and incompressible fluids. The proposed algorithm relies on the reformulation of the time‐discrete problem as a generalized Stokes problem with strongly variable coefficients, for which optimal preconditioners have recently been developed. The resulting algorithm is scalable, optimal, and robust: we test our implementation on model problems that mimic classical Turek–Hron benchmarks in two and three dimensions, and investigate timing and scalability results. [ABSTRACT FROM AUTHOR]

This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin (HDG) method. Modern implementations of high-order methods with state-of-the-art multigrid solvers for the Poisson equation are considered, including fast matrix-free implementations with sum factorization on quadrilateral and hexahedral elements. For the HDG method, a multigrid approach that combines a grid transfer from the trace space to the space of linear finite elements with algebraic multigrid on further levels is developed. It is found that high-order continuous finite elements give best time to solution for smooth solutions, closely followed by the matrix-free solvers for the other two discretizations. Their performance is up to an order of magnitude higher than that of the best matrix-based methods, even after including the superconvergence effects in the matrix-based HDG method. This difference is because of the vastly better performance of matrix-free operator evaluation as compared to sparse matrix-vector products. A rooine performance model confirms the superiority of the matrix-free implementation. [ABSTRACT FROM AUTHOR]

We develop a new matrix-free time-domain method, which requires no matrix solution, in unstructured meshes for general 3-D electromagnetic analysis. The method handles arbitrary unstructured meshes with the same ease as a finite-element method. Meanwhile, it is free of matrix solutions manifested by a naturally diagonal mass matrix, just like a finite-difference time-domain method. Different from our previous formulation where traditional curl-conforming vector bases are employed, modified vector bases are developed in this paper to directly connect the unknown coefficients of the vector basis functions employed to represent E (or H) with the unknowns obtained from the curl of H (or E), without any need for transformation. The proposed method employs only a single mesh. It does not require any interpolation and projection to obtain one field unknown from the other. Its accuracy and stability are guaranteed theoretically. Numerous experiments on unstructured triangular prism and tetrahedral meshes, involving both homogeneous and inhomogeneous and lossy materials, demonstrate the generality, accuracy, stability, and computational efficiency of the proposed method. The modified higher order vector bases developed in this paper can also be used in any other method that employs higher order bases to obtain an explicit relationship between unknown fields and unknown coefficients of vector bases. [ABSTRACT FROM PUBLISHER]

In this paper we consider a matrix-free path following algorithm for nonlinear parameter-dependent compact fixed point problems. We show that if these problems are discretized so that certain collective compactness and strong convergence properties hold, then this algorithm can follow smooth folds and capture simple bifurcations in a mesh-independent way. [ABSTRACT FROM AUTHOR]