Volume 16, Issue 4

We consider a non-isothermal multi-phase field model. We subsequently discretize implicitly in time and with linear finite elements. The arising algebraic problem is formulated in two variables where one is the multi-phase field, and the other contains the inverse temperature field. We solve this saddle point problem numerically by a non-smooth Schur–Newton approach using truncated non-smooth Newton multigrid methods. An application in grain growth as occurring in liquid phase crystallization of silicon is considered.

In this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of parabolic equations. Our finite point method has been tailored based on the local exponential basis functions. By the idea of our TFPM, we can recover all the traditional finite difference schemes. We can also construct some new TFPM schemes with better stability condition and accuracy. Furthermore, combining with the Shishkin mesh technique, we construct the uniformly convergent TFPM scheme for the convection-dominant convection-diffusion problem. Our numerical examples show the efficiency and reliability of TFPM.

We review the derivation of weakly penalized discontinuous Galerkin methods for scattering dominated radiation transport and extend the asymptotic analysis to non-isotropic scattering. We focus on the influence of the penalty parameter on the edges and derive a new penalty for interior edges and boundary fluxes. We study how the choice of the penalty parameters influences discretization accuracy and solver speed.

We present a new approach to the numerical upscaling for elliptic problems with rough diffusion coefficient at high contrast. It is based on the localizable orthogonal decomposition of H1${H^{1}}$ into the image and the kernel of some novel stable quasi-interpolation operators with local L2$L^{2}$-approximation properties, independent of the contrast. We identify a set of sufficient assumptions on these quasi-interpolation operators that guarantee in principle optimal convergence without pre-asymptotic effects for high-contrast coefficients. We then give an example of a suitable operator and establish the assumptions for a particular class of high-contrast coefficients. So far this is not possible without any pre-asymptotic effects, but the optimal convergence is independent of the contrast and the asymptotic range is largely improved over other discretization schemes. The new framework is sufficiently flexible to allow also for other choices of quasi-interpolation operators and the potential for fully robust numerical upscaling at high contrast.

This short article is the epilog of the 14 preceding papers in this and the previous issue of CMAM. All are extracted from the 5th Chinese–German Workshop on Computational and Applied Mathematics at Augsburg but submitted as individual papers to the journal.

The results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y , respectively. Our first main result is an error equality for all equations of the class A*⁢A⁢x+x=f${\mathrm{A}^{*}\mathrm{A}x+x=f}$ or in mixed formulation A*⁢y+x=f${\mathrm{A}^{*}y+x=f}$, A⁢x=y${\mathrm{A}x=y}$, where the exact solution (x,y)$(x,y)$ is in D⁢(A)×D⁢(A*)$D(\mathrm{A})\times D(\mathrm{A}^{*})$. Here A${\mathrm{A}}$ is a linear, densely defined and closed (usually a differential) operator and A*${\mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation (x~,y~)${(\tilde{x},\tilde{y})}$ belongs to D⁢(A)×D⁢(A*)${D(\mathrm{A})\times D(\mathrm{A}^{*})}$. In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality |x-x~|2+|A⁢(x-x~)|2+|y-y~|2+|A*⁢(y-y~)|2=ℳ⁢(x~,y~),$\lvert x-\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-% \tilde{y}\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}=\mathcal{M}(% \tilde{x},\tilde{y}),$ where ℳ⁢(x~,y~):=|f-x~-A*⁢y~|2+|y~-A⁢x~|2${\mathcal{M}(\tilde{x},\tilde{y}):=\lvert f-\tilde{x}-\mathrm{A}^{*}\tilde{y}% \rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. Our second main result is an error estimate for all equations of the class A*⁢A⁢x+i⁢x=f${\mathrm{A}^{*}\mathrm{A}x+ix=f}$ or in mixed formulation A*⁢y+i⁢x=f${\mathrm{A}^{*}y+ix=f}$, A⁢x=y${\mathrm{A}x=y}$, where i is the imaginary unit. For this problem we have the two-sided estimate 22+1⁢ℳi⁢(x~,y~)≤|x-x~|2+|A⁢(x-x~)|2+|y-y~|2+|A*⁢(y-y~)|2≤22-1⁢ℳi⁢(x~,y~),$\frac{\sqrt{2}}{\sqrt{2}+1}\mathcal{M}_{i}(\tilde{x},\tilde{y})\leq\lvert x-% \tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-\tilde{y}% \rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}\leq\frac{\sqrt{2}}{% \sqrt{2}-1}\mathcal{M}_{i}(\tilde{x},\tilde{y}),$ where ℳi⁢(x~,y~):=|f-i⁢x~-A*⁢y~|2+|y~-A⁢x~|2${\mathcal{M}_{i}(\tilde{x},\tilde{y}):=\lvert f-i\tilde{x}-\mathrm{A}^{*}% \tilde{y}\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.

A new algorithm for eigenvalue problems for linear differential operators with fractional derivatives is proposed and justified. The algorithm is based on the approximation (perturbation) of the coefficients of a part of the differential operator by piecewise constant functions where the eigenvalue problem for the last one is supposed to be simpler than the original one. Another milestone of the algorithm is the homotopy idea which results at the possibility for a given eigenpair number to compute recursively a sequence of the approximate eigenpairs. This sequence converges to the exact eigenpair with a super-exponential convergence rate. The eigenpairs can be computed in parallel for all prescribed indexes. The proposed method possesses the following principal property: its convergence rate increases together with the index of the eigenpair. Numerical examples confirm the theory.

The article deals with the analysis of a nonconforming finite element method for the discretization of optimization problems governed by variational inequalities. The state and adjoint variables are discretized using Crouzeix–Raviart nonconforming finite elements, and the control is discretized using a variational discretization approach. Error estimates have been established for the state and control variables. The results of numerical experiments are presented.

We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart–Thomas, and Brezzi–Douglas–Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.

We consider an optimal control problem subject to the thin-film equation. The PDE constraint lacks well-posedness for general right-hand sides due to possible degeneracies; state constraints are used to circumvent this problematic issue and to ensure well-posedness. Necessary optimality conditions for the optimal control problem are then derived. A convergent multi-parameter regularization is considered which addresses both, the possibly degenerate term in the equation and the state constraint. Some computational studies are then reported which evidence the relevant role of the state constraint, and motivate proper scalings of involved regularization and numerical parameters.