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The width of a well partial ordering (wpo) is the ordinal rank of the set of its antichains ordered by inclusion. We compute the width of wpos obtained as cartesian products of finitely many well-orderings.

We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilised) linearised discrete problems, and tends to a local minimum of the underlying energy functional. Simultaneously, the finite-dimensional approximation spaces are adaptively refined; this is implemented in terms of a new mesh refinement strategy in the context of finite element discretisations, which again relies on the energy structure of the problem under consideration, and does not involve any a posteriori error indicators. In combination, the resulting adaptive algorithm consists of an iterative linearisation procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge towards a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples.

In $d$ dimensions, approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$) suffers from the pollution effect if, as $k\to \infty$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the $h$-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$ fixed) suffers from the pollution effect, the celebrated papers [Melenk, Sauter 2010], [Melenk, Sauter 2011], [Esterhazy, Melenk 2012], and [Melenk, Parsania, Sauter 2013] showed that the $hp$-FEM (where accuracy is increased by decreasing the meshwidth $h$ and increasing the polynomial degree $p$) applied to a variety of constant-coefficient Helmholtz problems does not suffer from the pollution effect. The heart of the proofs of these results is a PDE result splitting the solution of the Helmholtz equation into "high" and "low" frequency components. The main novelty of the present paper is that we prove this splitting for the constant-coefficient Helmholtz equation in full-space (i.e., in $\mathbb{R}^d$) using only integration by parts and elementary properties of the Fourier transform (this is contrast to the proof for this set-up in [Melenk, Sauter 2010] which uses somewhat-involved bounds on Bessel and Hankel functions). We combine this splitting with (i) standard arguments about convergence of the FEM applied to the Helmholtz equation (the so-called "Schatz argument", which we reproduce here) and (ii) polynomial-approximation results (which we quote from the literature without proof) to give a simple proof that the $hp$-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation.

This paper proposes a regularization of the Monge-Amp\`ere equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge-Amp\`ere equation as the regularization parameter $\varepsilon$ approaches $0$. A mixed finite element method for the approximation of $u_\varepsilon$ is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.

We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to the second-order elliptic PDE in non-divergence form. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the 2D and 3D settings. In addition, we present a comparison with the existing multivariate spline methods in \cite{ALW06} and \cite{LW17} to show that the new method produces a similar and sometimes more accurate approximation in a more efficient fashion.

We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points/lines, where three interfaces meet, and at the boundary points/lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies.

We consider constrained partial differential equations of hyperbolic type with a small parameter $\varepsilon>0$, which turn parabolic in the limit case, i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is discussed and the corresponding solutions are compared in terms of the parameter $\varepsilon$. For the analysis, we consider the system equations as partial differential-algebraic equation based on the variational formulation of the problem. For a particular choice of the initial data, we reach first- and second-order estimates. For general initial data, lower-order estimates are proven and their optimality is shown numerically.

The aim of this work is to devise and analyse an accurate numerical scheme to solve Erd\'elyi-Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion and grey Brownian motion. To obtain convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erd\'elyi-Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singular in time term appearing in the main equation the proposed method converges slower than first order. Finally, we provide the numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions.

Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming $P_1$-$P_0$ element for the Stokes equation in three dimensions are constructed. And commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order $\boldsymbol H(\textrm{grad}\textrm{curl})$-nonconforming finite element only has $14$ degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The $\boldsymbol H(\textrm{grad}\textrm{curl})$-nonconforming elements are applied to solve the quad-curl problem, and optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming $P_1$-$P_0$ element method for the Stokes equation, based on which a fast solver is discussed. Numerical results are provided to verify the theoretical convergence rates.

This paper studies the inverse problem of determination the history for a stochastic diffusion process, by means of the value at the final time $T$. By establishing a new Carleman estimate, the conditional stability of the problem is proven. Based on the idea of Tikhonov method, a regularized solution is proposed. The analysis of the existence and uniqueness of the regularized solution, and proof for error estimate under an a-proior assumption are present. Numerical verification of the regularization, including numerical algorithm and examples are also illustrated.