Abstract This paper provides an overview of the new features of the finite element library deal.II , version 9.8, with substantially expanded support for mixed meshes containing tetrahedra, wedges, and pyramids, major improvements to the matrix-free GPU functionality, and many other improvements.
Abstract In this work, we consider a linearized model for fluid flow through a deformable porous elastic solid derived using mixture theory. The model accounts for small but non-negligible solid velocity relative to the fluid, resulting in a coupled system of equations. We focus on the error analysis and implementation of two fully discrete schemes: one using the first-order backward Euler method for time discretization and the other employing the second-order BDF2 scheme. We derive error estimates for both procedures and describe numerical experiments on the corresponding non-dimensional system.
Abstract We propose, analyze, and test an efficient splitting iteration for solving the incompressible, steady Navier–Stokes equations in the setting where partial solution data is known. The (possibly noisy) solution data is incorporated into a Picard-type solver via continuous data assimilation (CDA). Efficiency is gained through an algebraic splitting of Yosida-type that produces easier linear solves, and accuracy/consistency is shown to be maintained through the use of an incremental pressure and grad-div stabilization. We prove that CDA scales the Lipschitz constant of the associated fixed point operator by H 1/2 , where H is the characteristic spacing of the known solution data. This implies that CDA accelerates an already converging solver (and the more data, the more acceleration) and enables convergence of solvers in parameter regimes where the usual solver would fail (and the more data, the larger the parameter regime where convergence is achieved). Numerical tests illustrate the theory on several benchmark test problems.
In this paper, we are concerned with a posteriori error analysis for fully discrete solutions for the pseudostress-velocity formulation of the time dependent Stokes problem including nonhomogeneous mixed boundary conditions. The pseudostress-velocity formulation of the Stokes problem allows Raviart-Thomas finite elements. Piecewise constant discontinuous Galerkin method in time is proposed so that mixed finite element spaces are permitted to change at different time levels. Space-time a posteriori error estimates are derived by exploiting the residual and error representation formula in combination with the Stokes dual problem. Based on these estimates, we propose an adaptive space-time algorithm (ASTA), which includes features such as a space-time mesh change indicator and the separation of space error indicators from time error indicators. Numerical results highlight the superior efficiency of ASTA compared to uniform meshes. Additionally, our algorithm is able to effectively capture the corner singularity of the pressure variable in the backward-facing step flow problem.
This paper presents fully-discrete error analysis of the dynamically regularized Lagrange multiplier method (DRLM), a novel stabilization approach for the incompressible Navier-Stokes equations that incorporates kinetic energy evolution with a squared Lagrange multiplier regularization term into the system. The first-order implicit-explicit temporal discretization is employed for the DRLM formulation, in combination with the Taylor-Hood finite elements for spatial approximation. We establish optimal H-1-norm error estimate for the velocity and L-2-norm error estimate for the pressure through a mathematical induction process. Specifically, we first derive suboptimal error estimates, which serves as a basis for bounding the numerical solutions via the inverse inequalities, and optimal convergence can then be obtained by repeating the similar arguments provided with the uniformly bounded numerical solutions. Furthermore, we prove optimal L-2-norm error estimate for the velocity by using the negative norm techniques. Finally, numerical experiments are given to verify the theoretical convergence rates.
