Volume 17, Issue 1

We consider difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters α, β and γ. By the method of energy inequalities, for the solution of the difference problem, we obtain a priori estimates, which imply the stability and convergence of these difference schemes. The obtained results are supported by the numerical calculations carried out for some test problems.

In this paper, we design two classes of stabilized mixed finite element methods for linear elasticity on simplicial grids. In the first class of elements, we use 𝑯⁢(div,Ω;𝕊)${\boldsymbol{H}(\operatorname{div},\Omega;\mathbb{S})}$-Pk${P_{k}}$ and 𝑳2⁢(Ω;ℝn)${\boldsymbol{L}^{2}(\Omega;\mathbb{R}^{n})}$-Pk-1${P_{k-1}}$ to approximate the stress and displacement spaces, respectively, for 1≤k≤n${1\leq k\leq n}$, and employ a stabilization technique in terms of the jump of the discrete displacement over the edges/faces of the triangulation under consideration; in the second class of elements, we use 𝑯01⁢(Ω;ℝn)${\boldsymbol{H}_{0}^{1}(\Omega;\mathbb{R}^{n})}$-Pk${P_{k}}$ to approximate the displacement space for 1≤k≤n${1\leq k\leq n}$, and adopt the stabilization technique suggested by Brezzi, Fortin, and Marini [19]. We establish the discrete inf-sup conditions, and consequently present the a priori error analysis for them. The main ingredient for the analysis are two special interpolation operators, which can be constructed using a crucial 𝑯⁢(div)${\boldsymbol{H}(\operatorname{div})}$ bubble function space of polynomials on each element. The feature of these methods is the low number of global degrees of freedom in the lowest order case. We present some numerical results to demonstrate the theoretical estimates.

We consider the time fractional wave equation with coefficient which contains the Dirac delta distribution. The existence of generalized solutions of this initial-boundary value problem is proved. An implicit finite difference scheme approximating the problem is developed and its stability is proved. Estimates for the rate of convergence in special discrete energetic Sobolev norms are obtained. A numerical example confirms the theoretical results.

In this paper we give a new proof of the L∞${L^{\infty}}$-error estimate for the finite element approximation of the Laplace–Beltrami equation with an additional lower order term on a surface. While the proof available in the literature uses the method of perturbed bilinear forms from Schatz and Wahlbin, we adapt Scott’s proof from an Euclidean setting to the surface case. Furthermore, in contrast to the literature we use an approximative Green’s function on the surface instead of an exact Green’s function which is obtained by lifting an Euclidean Green’s function locally from the tangent plane to the surface.

We investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept. We investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the L2${L^{2}}$ sense, is also pointwise superconvergent in the temporal nodes. We prove a priori error estimates and we present numerical experiments to empirically support our findings.

The paper proves the convergence of the finite volume approximate solution of a convection-diffusion equation with an L1${L^{1}}$ right-hand side to the unique renormalized solution. The main difficulties are to handle the noncoercive character of the operator and the L1${L^{1}}$ data. Mixing the techniques of renormalized solutions and the finite volume method allows one to derive estimates for the discrete solutions and in particular a discrete version of the decay of the truncated energy. Indeed, as in the continuous case, the decay of the truncated energy is crucial to show that the limit of the approximate solution is the renormalized solution.

In this work, we propose and analyze an adaptive finite element method for a steady-state diffusion equation with a nonlinear boundary condition arising in cathodic protection. Under a general assumption on the marking strategy, we show that the algorithm generates a sequence of discrete solutions that converges strongly to the exact solution in H1⁢(Ω)${H^{1}(\Omega)}$ and the sequence of error estimators has a vanishing limit. Numerical results show clearly the convergence and efficiency of the adaptive algorithm.

An efficient algorithm is proposed and studied for computing flow ensembles of incompressible magnetohydrodynamic (MHD) flows under uncertainties in initial or boundary data. The ensemble average of J realizations is approximated through a clever algorithm (adapted from a breakthrough idea of Jiang and Layton [23]) that, at each time step, uses the same matrix for each of the J systems solves. Hence, preconditioners need to be built only once per time step, and the algorithm can take advantage of block linear solvers. Additionally, an Elsässer variable formulation is used, which allows for a stable decoupling of each MHD system at each time step. We prove stability and convergence of the algorithm, and test it with two numerical experiments.

In this paper, quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides are considered, with the quadrature methods generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, and the smoothness of the involved functions is carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a version which sometimes requires less amount of computational work than the standard version of this principle. Numerical results are included.

This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.