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.
This study introduces a hybridizable discontinuous Galerkin (HDG) method for simulating low-frequency wave propagation in poroelastic media. We present a novel four-field variational formulation and establish its well-posedness and energy stability. Our hp-convergence analysis of the HDG method for spatial discretization is complemented by a Crank-Nicolson scheme for temporal discretization. Numerical experiments validate the theoretical convergence rates and demonstrate the effectiveness of the method in accurately capturing poroelastic dynamics.
In the context of numerical methods for time-domain wave propagation problems, combining high-order lumped finite elements with an explicit time scheme is a popular approach for either inviscid or visco-elastic models. This strategy has proven to be efficient in numerous cases. However, when dealing with non-uniform meshes or high-contrast materials, the stability condition on the time step becomes drastically stringent. One can encounter such configurations when meshing "unfortunate" CAD input file, e.g. when dealing with heterogeneous materials where neighboring heterogeneities produce very small elements in-between them, or when considering materials with high and localized wave velocities. To address efficiently these configurations, we propose to adapt the locally implicit and stabilized explicit methods to the Kelvin-Voigt, Maxwell and Zener visco-elastic models. We prove using energy arguments that the global stability condition of these schemes can be much more favorable compared to a leapfrog explicit scheme, decreasing the number of iterations for a fixed time window. We illustrate our approaches with 2D and 3D numerical test cases related to ultrasonic non-destructive testing experiments.
Sequences of block matrices with rectangular Toeplitz blocks arise in several applications, including the numerical discretization of differential problems. We compute the (asymptotic) spectral and singular value distribution of these sequences in the case where the asymptotic block size ratios are not necessarily rational as in [A. Adriani, I. Furci, C. Garoni, and S. Serra-Capizzano, Spectral and singular value distribution of sequences of block matrices with rectangular Toeplitz blocks. Part I: Asymptotically rational block size ratios. J. Numer. Math. DOI: 10.1515/jnma-2025-0091]. Our derivation relies on the recent notion of generalized approximating classes of sequences as well as on a functional-based approach to describe the spectral and singular value distributions. The distribution result is illustrated through numerical experiments, one of which is inspired by the finite element approximation of a system of differential equations.
Motivated by several applications, especially in the context of numerical discretizations of differential problems, we compute the (asymptotic) spectral and singular value distribution of sequences of block matrices formed by rectangular Toeplitz blocks. We remark that this computation cannot be achieved through the theory of generalized locally Toeplitz (GLT) sequences whenever the Toeplitz blocks have different sizes. In this sense, the present paper paves the way for an enhancement of the GLT apparatus in order to manage block structures formed by rectangular blocks of different sizes. The distribution result is illustrated through numerical experiments, one of which is inspired by the numerical discretization of an interface problem.
A decoupled finite element method based on the scalar auxiliary variable and vector penalty projection approach is constructed and analysed for solving a fluid-fluid interaction problem, which includes two Navier-Stokes equations coupled by some nonlinear interface conditions. The proposed full-discrete scheme is a combination of mixed finite element approximation for spatial discretization, backward Euler scheme for temporal discretization, as well as explicit treatment for the interface conditions, and can penalize for lack of mass conservation. Furthermore, unconditional energy stability is given and error estimates for the fully discrete scheme are showed. Finally, some numerical experiments are provided to illustrate the theoretical results and efficiency of the presented method.
Continuous data assimilation addresses time-dependent problems with unknown initial conditions by incorporating observations of the solution into a nudging term. For the prototypical heat equation with variable conductivity and the Neumann boundary condition, we consider data assimilation schemes with non-interpolant observables unlike previous studies. These generalized nudging strategies are notably useful for problems which possess limited or even no additional regularity beyond the minimal framework. We demonstrate that a spatially discretized nudged solution converges exponentially fast in time to the true solution with the rate guaranteed by the choice of the nudging strategy independent of the discretization. Furthermore, the long-term discrete error is optimal as it matches the estimates available for problems of limited regularity with known initial conditions. Three particular strategies -- nudging by a conforming finite element subspace, nudging by piecewise constants on the boundary mesh, and nudging by the mean value -- are explored numerically for three test cases, including a problem with Dirac delta forcing and the Kellogg problem with discontinuous conductivity.
In this paper, we develop a novel staggered mesh (SM) approach for general nonlinear dissipative systems with arbitrary energy distributions (including cases with known or unknown energy lower bounds). Based on this framework, we propose several second-order semi-discrete schemes that maintain linearity, computational decoupling, and unconditional energy stability. Firstly, for dissipative systems with known energy lower bounds, we introduce a positive auxiliary variable V(t) to substitute the total energy functional, subsequently discretizing it on staggered temporal meshes to ensure that the energy remains non-increasing regardless of the size of time step. The newly developed schemes achieve full computational decoupling, maintaining essentially the same computational expense as conventional implicit-explicit methods while demonstrating significantly improved accuracy. Furthermore, we rigorously establish the positivity preservation of the discrete variable V^n+1/2 which is a crucial property ensuring numerical stability and accuracy. Theoretical analysis confirms second-order temporal convergence for the proposed SM schemes. Secondly, for dissipative systems lacking well-defined energy lower bounds, we devise an alternative auxiliary variable formulation and extend the SM framework to maintain unconditional energy stability while preserving numerical effectiveness and accuracy. Finally, comprehensive numerical experiments, including benchmark problem simulations, validate the proposed schemes' efficacy and demonstrate their superior performance characteristics.
In this article, a hybridizable discontinuous Galerkin (HDG) method is proposed and analyzed for the Klein-Gordon equation with local Lipschitz-type non-linearity. A priori error estimates are derived, and it is proved that approximations of the flux and the displacement converge with order O(h k+1), where h is the discretizing parameter and k is the degree of the piecewise polynomials to approximate both flux and displacement variables. After post-processing of the semi-discrete solution, it is shown that the post-processed solution converges with order O(h k+2) for k >= 1. Moreover, a second-order conservative finite difference scheme is applied to discretize in time and it is proved that the discrete energy is conserved with optimal error estimates for the completely discrete method. To avoid solving a nonlinear system of algebraic equations at each time step, a non-conservative scheme is proposed, and its error analysis is also briefly established. Moreover, another variant of the HDG scheme is analyzed, and error estimates are established. Finally, some numerical experiments are conducted to confirm our theoretical findings.
We consider a scalar conservation law with linear and nonlinear flux function on a bounded domain Omega subset of R 2 ${\Omega}\subset {\mathbb{R}}<^>{2}$ with Lipschitz boundary partial derivative Omega. We discretize the spatial variable with the standard finite element method, where we use a local extremum diminishing flux limiter, which is linearity preserving. For temporal discretization, we use the second order explicit strong stability preserving Runge-Kutta method. It is known that the resulting fully-discrete scheme satisfies the discrete maximum principle. Under the sufficiently regularity of the weak solution and the CFL condition k = O ( h 2 ) $k=\mathcal{O}\left({h}<^>{2}\right)$ , we derive error estimates in & ell; infinity(L 2)-norm for the algebraic flux correction scheme. We also present numerical experiments that validate that the fully-discrete scheme satisfies the temporal order of convergence of the fully-discrete scheme that we proved in the theoretical analysis.
In this paper we analyze a space-time unfitted finite element method for the discretization of scalar surface partial differential equations on evolving surfaces. For higher order approximations of the evolving surface we use the technique of (iso)parametric mappings for which a level set representation of the evolving surface is essential. We derive basic results in which certain geometric characteristics of the exact space-time surface are related to corresponding ones of the numerical surface approximation. These results are used in an error analysis of a higher order space-time TraceFEM.
Trace finite element methods have become a popular option for solving surface partial differential equations, especially in problems where surface and bulk effects are coupled. In such methods a surface mesh is formed by approximately intersecting the continuous surface on which the PDE is posed with a three-dimensional (bulk) tetrahedral mesh. In classical $H^1$-conforming trace methods, the surface finite element space is obtained by restricting a bulk finite element space to the surface mesh. It is not clear how to carry out a similar procedure in order to obtain other important types of finite element spaces such as $H({\rm div})$-conforming spaces. Following previous work of Olshanskii, Reusken, and Xu on $H^1$-conforming methods, we develop a ``quasi-trace'' mixed method for the Laplace-Beltrami problem. The finite element mesh is taken to be the intersection of the surface with a regular tetrahedral bulk mesh as previously described, resulting in a surface triangulation that is highly unstructured and anisotropic but satisfies a classical maximum angle condition. The mixed method is then employed on this mesh. Optimal error estimates with respect to the bulk mesh size are proved along with superconvergent estimates for the projection of the scalar error and a postprocessed scalar approximation.
We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which, we show that it is possible to simulate the Fr\'eedericksz transition with this scheme.
Mixed methods for linear elasticity with strongly symmetric stresses of lowest order are studied in this paper. On each simplex, the stress space has piecewise linear components with respect to its Alfeld split (which connects the vertices to barycenter), generalizing the Johnson–Mercier two-dimensional element to higher dimensions. Further reductions in the stress space in the three-dimensional case (to 24 degrees of freedom per tetrahedron) are possible when the displacement space is reduced to local rigid displacements. Proofs of optimal error estimates of numerical solutions and improved error estimates via postprocessing and the duality argument are presented.
This paper provides an overview of the new features of the finite element library deal.II, version 9.7.
We describe a representation of C 1 piecewise polynomials defined using generalized Hermite elements in two dimensions. This provides a way to generate C 1 piecewise polynomials using nodal variables that are the same at all vertices for degree n ⩾ 5. From this, one obtains both the dimension of the space and an interpolant.
We present an approach to shape optimization problems that uses an unfitted finite element method (FEM). The domain geometry is represented, and optimized, using a (discrete) level set function and we consider objective functionals that are defined over bulk domains. For a discrete objective functional, defined in the unfitted FEM framework, we show that the exact discrete shape derivative essentially matches the shape derivative at the continuous level. In other words, our approach has the benefits of both optimize-then-discretize and discretize-then-optimize approaches. Specifically, we establish the shape Fréchet differentiability of discrete (unfitted) bulk shape functionals using both the perturbation of the identity approach and direct perturbation of the level set representation. The latter approach is especially convenient for optimizing with respect to level set functions. Moreover, our Fréchet differentiability results hold for any polynomial degree used for the discrete level set representation of the domain. We illustrate our results with some numerical accuracy tests, a simple model (geometric) problem with known exact solution, as well as shape optimization of structural designs.
