Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
July 17, 2021
### Abstract

We analyze backward Euler time stepping schemes for a primal DPG formulation of a class of parabolic problems. Optimal error estimates are shown in a natural norm and in the L2{L^{2}} norm of the field variable. For the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature. In the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators. It is essential that these operators be projections with respect to the spatial part of the PDE, as in standard Galerkin schemes, and not with respect to the full PDE at a time step, as done previously.

Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
July 17, 2021
### Abstract

Time-fractional initial-boundary problems of parabolic type are considered. Previously, global error bounds for computed numerical solutions to such problems have been provided by Liao et al. (SIAM J. Numer. Anal. 2018, 2019) and Stynes et al. (SIAM J. Numer. Anal. 2017). In the present work we show how the concept of complete monotonicity can be combined with these older analyses to derive local error bounds (i.e., error bounds that are sharper than global bounds when one is not close to the initial time t=0{t=0}). Furthermore, we show that the error analyses of the above papers are essentially the same – their key stability parameters, which seem superficially different from each other, become identical after a simple rescaling. Our new approach is used to bound the global and local errors in the numerical solution of a multi-term time-fractional diffusion equation, using the L1 scheme for the temporal discretisation of each fractional derivative. These error bounds are α-robust. Numerical results show they are sharp.

Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
June 29, 2021
### Abstract

In this work, we develop novel adaptive hybrid discontinuous Galerkin algorithms for second-order elliptic problems. For this, two types of reliable and efficient, modulo a data-oscillation term, and fully computable a posteriori error estimators are developed: the first one is a simple residual type error estimator, and the second is a flux reconstruction based error estimator of a guaranteed type for polynomial approximations of any degree by using a simple postprocessing. These estimators can achieve high-order accuracy for both smooth and nonsmooth problems even with high-order approximations. In order to enhance the performance of adaptive algorithms, we introduce 𝐾-means clustering based marking strategy. The choice of marking parameter is crucial in the performance of the existing strategy such as maximum and bulk criteria; however, the optimal choice is not known. The new strategy has no unknown parameter. Several numerical examples are given to illustrate the performance of the new marking strategy along with our estimators.

Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
June 9, 2021
### Abstract

The goal of this paper is to present various types of iterative solvers: gradient iteration, Newton’s method and a quasi-Newton method, for the finite element solution of elliptic problems arising in Gao type beam models (a geometrical type of nonlinearity, with respect to the Euler–Bernoulli hypothesis). Robust behaviour, i.e., convergence independently of the mesh parameters, is proved for these methods, and they are also tested with numerical experiments.

Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
June 5, 2021
### Abstract

This study investigates the dual system least-squares finite element method, namely the LL ∗ method, for a hyperbolic problem. It mainly considers nonlinear hyperbolic conservation laws and proposes a combination of the LL ∗ method and Newton’s iterative method. In addition, the inclusion of a stabilizing term in the discrete LL ∗ minimization problem is proposed, which has not been investigated previously. The proposed approach is validated using the one-dimensional Burgers equation, and the numerical results show that this approach is effective in capturing shocks and provides approximations with reduced oscillations in the presence of shocks.

Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
April 30, 2021
### Abstract

A time-fractional initial-boundary value problem of wave type is considered, where the spatial domain is (0,1)d(0,1)^{d} for some d∈{1,2,3}d\in\{1,2,3\}. Regularity of the solution 𝑢 is discussed in detail. Typical solutions have a weak singularity at the initial time t=0t=0: while 𝑢 and utu_{t} are continuous at t=0t=0, the second-order derivative uttu_{tt} blows up at t=0t=0. To solve the problem numerically, a finite difference scheme is used on a mesh that is graded in time and uniform in space with the same mesh size ℎ in each coordinate direction. This scheme is generated through order reduction: one rewrites the differential equation as a system of two equations using the new variable v:=utv:=u_{t}; then one uses a modified L1 scheme of Crank–Nicolson type for the driving equation. A fast variant of this finite difference scheme is also considered, using a sum-of-exponentials (SOE) approximation for the kernel function in the Caputo derivative. The stability and convergence of both difference schemes are analysed in detail. At each time level, the system of linear equations generated by the difference schemes is solved by a fast Poisson solver, thereby taking advantage of the fast difference scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of both numerical methods.

Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
April 20, 2021
### Abstract

We consider systems of ordinary differential equations with multiple scales in time. In general, we are interested in the long time horizon of a slow variable that is coupled to solution components that act on a fast scale. Although the fast scale variables are essential for the dynamics of the coupled problem, they are often of no interest in themselves. Recently, we have proposed a temporal multiscale approach that fits into the framework of the heterogeneous multiscale method and that allows for efficient simulations with significant speedups. Fast and slow scales are decoupled by introducing local averages and by replacing fast scale contributions by localized periodic-in-time problems. Here, we generalize this multiscale approach to a larger class of problems, but in particular, we derive an a posteriori error estimator based on the dual weighted residual method that allows for a splitting of the error into averaging error, error on the slow scale and error on the fast scale. We demonstrate the accuracy of the error estimator and also its use for adaptive control of a numerical multiscale scheme.

Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
April 2, 2021
### Abstract

A gradient discretisation method (GDM) is an abstract setting that designs the unified convergence analysis of several numerical methods for partial differential equations and their corresponding models. In this paper, we study the GDM for anisotropic reaction–diffusion problems, based on a general reaction term, with Neumann boundary condition. With natural regularity assumptions on the exact solution, the framework enables us to provide proof of the existence of weak solutions for the problem, and to obtain a uniform-in-time convergence for the discrete solution and a strong convergence for its discrete gradient. It also allows us to apply non-conforming numerical schemes to the model on a generic grid (the non-conforming ℙ1{\mathbb{P}1} finite element scheme and the hybrid mixed mimetic (HMM) methods). Numerical experiments using the HMM method are performed to assess the accuracy of the proposed scheme and to study the growth of glioma tumors in heterogeneous brain environment. The dynamics of their highly diffusive nature is also measured using the fraction anisotropic measure. The validity of the HMM is examined further using four different mesh types. The results indicate that the dynamics of the brain tumor is still captured by the HMM scheme, even in the event of a highly heterogeneous anisotropic case performed on the mesh with extreme distortions.

Unable to retrieve citations for this document

Retrieving citations for document...

Open Access
March 2, 2021
### Abstract

We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
February 25, 2021
### Abstract

In this paper, we consider the problem of reconstructing a space-dependent coefficient in a linear Benjamin–Bona–Mahony (BBM)-type equation from a single measurement of its solution at a given time. We analyze the well-posedness of the forward initial-boundary value problem and characterize the inverse problem as a constrained optimization one. Our objective consists on reconstructing the variable coefficient in the BBM equation by minimizing an appropriate regularized Tikhonov-type functional constrained by the BBM equation. The well-posedness of the forward problem is studied and approximated numerically by combining a finite-element strategy for spatial discretization using the Python-FEniCS package, together with a second-order implicit scheme for time stepping. The minimization process of the Tikhonov-regularization adopted is performed by using an iterative L-BFGS-B quasi-Newton algorithm as described for instance by Byrd et al. (1995) and Zhu et al. (1997). Numerical simulations are presented to demonstrate the robustness of the proposed method with noisy data. The local stability and uniqueness of the solution to the constrained optimization problem for a fixed value of the regularization parameter are also proved and illustrated numerically.

Unable to retrieve citations for this document

Retrieving citations for document...

Requires Authentication
Accessible
February 20, 2021
### Abstract

In this paper, we propose an improvement of the classical compact finite difference (CFD) method by using a proper orthogonal decomposition (POD) technique for time-fractional diffusion equations in one- and two-dimensional space. A reduced CFD method is constructed with lower dimensions such that it maintains the accuracy and decreases the computational time in comparison with classical CFD method. Since the solution of time-fractional diffusion equation typically has a weak singularity near the initial time t=0{t=0}, the classical L1 scheme on uniform meshes may obtain a scheme with low accuracy. So, we use the L1 scheme on graded meshes for time discretization. Moreover, we provide the error estimation between the reduced CFD method based on POD and classical CFD solutions. Some numerical examples show the effectiveness and accuracy of the proposed method.

Unable to retrieve citations for this document

Retrieving citations for document...

Accessible
February 2, 2021
### Abstract

In this paper, based on the shift splitting technique, a shift splitting (SS) iteration method is presented to solve the generalized absolute value equations. Convergence conditions of the SS method are discussed in detail when the involved matrices are some special matrices. Finally, numerical experiments show the effectiveness of the proposed method.

Unable to retrieve citations for this document

Retrieving citations for document...

Accessible
December 11, 2020
### Abstract

In this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence rate when the time discretization is performed on uniform meshes. Therefore, in order to improve the convergence order, the Caputo time fractional derivative term is discretized by the L2-1σ{L2-1_{\sigma}} format on non-uniform meshes, with σ=1-α2{\sigma=1-\frac{\alpha}{2}}, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. According to the summation formula of positive integer k power, and considering k=3,4,5{k=3,4,5}, we propose three non-uniform meshes for time discretization. Through theoretical analysis, different time convergence orders O(N-min{kα,2}){O(N^{-\min\{k\alpha,2\}})} can be obtained, where N denotes the number of time splits. Finally, the theoretical analysis is verified by several numerical examples.