Volume 15, Issue 4

We consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We complement in a number of ways earlier studies of the possible extension of this fact to spatially semidiscrete and fully discrete piecewise linear finite element discretizations, based on the standard Galerkin method, the lumped mass method, and the finite volume element method. We also provide numerical examples that illustrate our findings.

We design an adaptive wavelet scheme for solving first-order system least-squares formulations of second-order elliptic PDEs that converge with the best possible rate in linear complexity. A wavelet Riesz basis is constructed for the space H→0,ΓN(div;Ω)$\vec{H}_{0,\Gamma _N}(\operatorname{div};\Omega )$ on general polygons. The theoretical findings are illustrated by numerical experiments.

In this paper, we present and analyze a FETI-DP solver with deluxe scaling for a Nitsche-type discretization [Comput. Methods Appl. Math. 3 (2003), 76–85], [SIAM J. Numer. Anal. 49 (2011), 1761–1787] based on a discontinuous Galerkin (DG) method for elliptic two-dimensional problems with discontinuous coefficients and non-matching meshes only across subdomains. We establish a condition number estimate for the preconditioned linear system which is scalable with respect to the number of subdomains, is quasi-optimal polylogarithmic with respect to subdomain mesh size, and is independent of coefficient discontinuities and ratio of mesh sizes across subdomain interfaces. Numerical experiments support the theory and show that the deluxe scaling improves significantly the performance over classical scaling.

We give an overview on existence results in nonlinear peridynamics, a nonlocal elasticity theory, and compare these results with those known for classical nonlinear elastodynamics. In peridynamics, it is possible to obtain stronger results under equal or even less restrictive assumptions on the stress.

We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order O(Δx3-α)${O(\Delta x^{3- \alpha })}$, 1<α<2${1&lt;\alpha &lt;2}$. A shifted implicit finite difference method is applied for solving the two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is O(Δt+Δxmin(3-α,β))${O (\Delta t + \Delta x^{\min (3- \alpha , \beta )})}$, 1<α<2${1&lt; \alpha &lt;2}$, β>0${\beta &gt;0}$, where Δt,Δx${\Delta t,\Delta x}$ denote the time and space stepsizes, respectively. Numerical examples where the solutions have varying degrees of smoothness are presented and compared with the exact analytical solution to compare the practical performance of the method with the theoretical order of convergence.

The paper is concerned with computable estimates of the distance between a vector-valued function in the Sobolev space W1,γ(Ω,ℝd)$W^{1,\gamma }(\Omega ,\mathbb {R}^d)$ (where γ∈(1,+∞)${\gamma \in (1,+\infty )}$ and Ω is a bounded Lipschitz domain in ℝ d ) and the subspace S1,γ(Ω,ℝd)${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$ containing all divergence-free (solenoidal) vector functions. Derivation of these estimates is closely related to the stability theorem that establishes existence of a bounded operator inverse to the operator div${\operatorname{div}}$. The constant in the respective stability inequality arises in the estimates of the distance to the set S1,γ(Ω,ℝd)${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$. In general, it is difficult to find a guaranteed and realistic upper bound of this global constant. We suggest a way to circumvent this difficulty by using weak (integral mean) solenoidality conditions and localized versions of the stability theorem. They are derived for the case where Ω is represented as a union of simple subdomains (overlapping or non-overlapping), for which estimates of the corresponding stability constants are known. These new versions of the stability theorem imply estimates of the distance to S1,γ(Ω,ℝd)${S^{1,\gamma }(\Omega ,\mathbb {R}^d)}$ that involve only local constants associated with subdomains. Finally, the estimates are used for deriving fully computable a posteriori estimates for problems in the theory of incompressible viscous fluids.

Several open questions in the numerical analysis of singularly perturbed differential equations are discussed. These include whether certain convergence results in various norms are optimal, when supercloseness is obtained in finite element solutions, the validity of defect correction in finite difference approximations, and desirable adaptive mesh refinement results that remain to be proved or disproved.

We propose and analyze a space-time finite element method for the numerical solution of parabolic evolution equations. This approach allows the use of general and unstructured space-time finite elements which do not require any tensor product structure. The stability of the numerical scheme is based on a stability condition which holds for standard finite element spaces. We also provide related a priori error estimates which are confirmed by numerical experiments.