A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods

In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.

Sufficient conditions for the existence of heteroclinic solutions for Phi-Laplacian differential equations

In this paper we consider the second order discontinuous equation in the real line, (a(t)φ(u′(t)))′ = f(t,u(t),u′(t)), a.e.t∈R, u(-∞) = ν⁻, u(+∞)=ν⁺, with φ an increasing homeomorphism such that φ(0)=0 and φ(R)=R, a∈C(R,R\{0})∩C¹(R,R) with a(t)>0, or a(t)<0, for t∈R, f:R³→R a L¹-Carathéodory function and ν⁻,ν⁺∈R such that ν⁻<ν⁺. We point out that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities φ and f. Moreover, as far as we know, this result is even new when φ(y)=y, that is, for equation (a(t)u′(t))′=f(t,u(t),u′(t)), a.e.t∈R.

On S-asymptotically omega-periodic functions on Banach spaces and applications

This paper is devoted to the study of the class of continuous and bounded functions f : [0, infinity] -> X for which exists omega > 0 such that lim(t ->infinity) (f (t + omega) - f (t)) = 0 (in the sequel called S-asymptotically omega-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically omega-periodic functions. We also study the existence of S-asymptotically omega-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces. (C) 2008 Elsevier Inc. All rights reserved.

Chebyshev real wave packet propagation: H+O-2 (J=0) state-to-state reactive scattering calculations

In this paper we explore the relative performance of two recently developed wave packet methodologies for reactive scattering, namely the real wave packet Chebyshev domain propagation of Gray and Balint-Kurti [J. Chem. Phys. 108, 950 (1998)] and the Lanczos subspace wave packet approach of Smith [J. Chem. Phys. 116, 2354 (2002); Chem. Phys. Lett. 336, 149 (2001)]. In the former method, a modified Schrodinger equation is employed to propagate the real part of the wave packet via the well-known Chebyshev iteration. While the time-dependent wave packet from the modified Schrodinger equation is different from that obtained using the standard Schrodinger equation, time-to-energy Fourier transformation yields wave functions which differ only trivially by normalization. In the Lanczos subspace approach the linear system of equations defining the action of the Green operator may be solved via either time-dependent or time-independent methods, both of which are extremely efficient due to the simple tridiagonal structure of the Hamiltonian in the Lanczos representation. The two different wave packet methods are applied to three dimensional reactive scattering of H+O-2 (total J=0). State-to-state reaction probabilities, product state distributions, as well as initial-state-resolved cumulative reaction probabilities are examined. (C) 2002 American Institute of Physics.

Numerical methods in multibody system dynamics

"Series: Solid mechanics and its applications, vol. 226"

Exercise sheet 7

Exercises and solutions in LaTex

Exercise sheet 7 pdf

Exercises and solutions in PDF

Impact of mass lumping on gravity and Rossby waves in 2D finite-element shallow-water models

The goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finite-element velocity/surface-elevation pairs that are used to approximate the linear shallow-water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P0-P1, RT0 and P-P1 pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

Generalised Dirichelt-to-Neumann map in time dependent domains

We study the heat, linear Schrodinger and linear KdV equations in the domain l(t) < x < ∞, 0 < t < T, with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.

The univalent polynomial of Suffridge as a summability kernel

A positive summability trigonometric kernel {K(n)(theta)}(infinity)(n=1) is generated through a sequence of univalent polynomials constructed by Suffridge. We prove that the convolution {K(n) * f} approximates every continuous 2 pi-periodic function f with the rate omega(f, 1/n), where omega(f, delta) denotes the modulus of continuity, and this provides a new proof of the classical Jackson`s theorem. Despite that it turns out that K(n)(theta) coincide with positive cosine polynomials generated by Fejer, our proof differs from others known in the literature.

Static Hopfions in the extended Skyrme-Faddeev model

We construct static soliton solutions with non-zero Hopf topological charges to a theory which is an extension of the Skyrme-Faddeev model by the addition of a further quartic term in derivatives. We use an axially symmetric ansatz based on toroidal coordinates, and solve the resulting two coupled non-linear partial differential equations in two variables by a successive over-relaxation (SOR) method. We construct numerical solutions with Hopf charge up to four, and calculate their analytical behavior in some limiting cases. The solutions present an interesting behavior under the changes of a special combination of the coupling constants of the quartic terms. Their energies and sizes tend to zero as that combination approaches a particular special value. We calculate the equivalent of the Vakulenko and Kapitanskii energy bound for the theory and find that it vanishes at that same special value of the coupling constants. In addition, the model presents an integrable sector with an in finite number of local conserved currents which apparently are not related to symmetries of the action. In the intersection of those two special sectors the theory possesses exact vortex solutions (static and time dependent) which were constructed in a previous paper by one of the authors. It is believed that such model describes some aspects of the low energy limit of the pure SU(2) Yang-Mills theory, and our results may be important in identifying important structures in that strong coupling regime.

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged. leads to a linear system of equations for the inter-face configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin`s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system`s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve. (C) 2009 Elsevier Inc. All rights reserved.

Hyperfunctions and (analytic) hypoellipticity

In this work we discuss the problem of smooth and analytic regularity for hyperfunction solutions to linear partial differential equations with analytic coefficients. In particular we show that some well known ""sum of squares"" operators, which satisfy Hormander`s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).

Estimativa da área foliar de Brachiaria plantaginea usando dimensões lineares do limbo foliar

Com o objetivo de obter uma equação que, por meio de parâmetros lineares dimensionais das folhas, permita a estimativa da área foliar de Brachiaria plantaginea, estudaram-se relações entre a área foliar real (Sf) e os parâmetros dimensionais do limbo foliar, como o comprimento ao longo da nervura principal (C) e a largura máxima (L), perpendicular à nervura principal. As equações lineares simples, exponenciais e geométricas obtidas podem ser usadas para estimação da área foliar do capim-marmelada. do ponto de vista prático, deve-se optar pela equação linear simples, envolvendo o produto C x L, usando-se a equação de regressão Sf = 0,7338 x (C x L), o que equivale a tomar 73,38% do produto entre o comprimento ao longo da nervura principal e a largura máxima, com um coeficiente de determinação de 0,8754.