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SIAM J. on Numerical Analysis

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2010

Volume 48, Issue 6, pp. 2019-2266


Explicit Runge–Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems

Erik Burman, Alexandre Ern, and Miguel A. Fernández

SIAM J. Numer. Anal. 48, pp. 2019-2042 (24 pages) | Cited 2 times

Online Publication Date: December 02, 2010

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We analyze explicit Runge–Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs type. For the time discretization, we consider explicit second- and third-order Runge–Kutta schemes. We identify a general set of properties on the space stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then we establish $L^2$-norm error estimates with quasi-optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for third-order Runge–Kutta schemes and any polynomial degree in space and for second-order Runge–Kutta schemes and first-order polynomials in space. For second-order Runge–Kutta schemes and higher polynomial degrees in space, a tightened 4/3-CFL condition is required. Numerical results are presented for smooth and rough solutions. The case of finite volumes is briefly discussed.

Computing Sacker–Sell spectra in Discrete Time Dynamical Systems

Thorsten Hüls

SIAM J. Numer. Anal. 48, pp. 2043-2064 (22 pages)

Online Publication Date: December 02, 2010

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In this paper we develop boundary value methods for detecting Sacker–Sell spectra in discrete time dynamical systems. The algorithms are advancements of earlier methods for computing projectors of exponential dichotomies. The first method is based on the projector residual $P^2-P$. If this residual is large, then the difference equation has no exponential dichotomy. Further criterions for detecting Sacker–Sell spectral intervals are the norm of end points and midpoints of the solution of a specific boundary value problem. Refined error estimates for the underlying approximation process are given, and the resulting algorithms are applied to an example with known continuous Sacker–Sell spectrum, as well as to the variational equation along orbits of Hénon's map.

Multiscale Analysis in Sobolev Spaces on the Sphere

Q. T. Le Gia, I. H. Sloan, and H. Wendland

SIAM J. Numer. Anal. 48, pp. 2065-2090 (26 pages)

Online Publication Date: December 02, 2010

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We consider a multiscale approximation scheme at scattered sites for functions in Sobolev spaces on the unit sphere $\mathbb{S}^n$. The approximation is constructed using a sequence of scaled, compactly supported radial basis functions restricted to $\mathbb{S}^n$. A convergence theorem for the scheme is proved, and the condition number of the linear system is shown to stay bounded by a constant from level to level, thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points.

Analysis and Optimization of Robin–Robin Partitioned Procedures in Fluid-Structure Interaction Problems

Luca Gerardo-Giorda, Fabio Nobile, and Christian Vergara

SIAM J. Numer. Anal. 48, pp. 2091-2116 (26 pages)

Online Publication Date: December 02, 2010

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In the solution of fluid-structure interaction (FSI) problems, partitioned procedures are modular algorithms that involve separate fluid and structure solvers that interact in an iterative framework through the exchange of suitable transmission conditions at the FS interface. In this work we study, using Fourier analysis, the convergence of partitioned algorithms based on Robin transmission conditions. We derive, for different models of the fluid and the structure, a frequency-dependent reduction factor at each iteration of the partitioned algorithm, which is minimized by choosing optimal values of the coefficients in the Robin transmission conditions. Two-dimensional numerical results are also reported, which highlight the effectiveness of the optimization procedure.

An Analysis of a Broken $P_1$-Nonconforming Finite Element Method for Interface Problems

Do Y. Kwak, Kye T. Wee, and Kwang S. Chang

SIAM J. Numer. Anal. 48, pp. 2117-2134 (18 pages)

Online Publication Date: December 08, 2010

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We study some numerical methods for solving a second order elliptic problem with interface. We introduce an immersed finite element method based on the “broken” $P_1$-nonconforming piecewise linear polynomials on interface triangular elements having edge averages as degrees of freedom. These linear polynomials are broken to match the homogeneous jump condition along the interface which is allowed to cut through the element. We prove optimal orders of convergence in the $H^1$- and $L^2$-norm. Next we propose a mixed finite volume method in the context introduced in [S. H. Chou, D. Y. Kwak, and K. Y. Kim, Math. Comp., 72 (2003), pp. 525–539] using the Raviart–Thomas mixed finite element and this “broken” $P_1$-nonconforming element. The advantage of this mixed finite volume method is that once we solve the symmetric positive definite pressure equation (without Lagrangian multiplier), the velocity can be computed locally by a simple formula. This procedure avoids solving the saddle point problem. Furthermore, we show optimal error estimates of velocity and pressure in our mixed finite volume method. Numerical results show optimal orders of error in the $L^2$-norm and broken $H^1$-norm for the pressure and in the $H(\mathrm{div})$-norm for the velocity.

Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

Congpei AN, Xiaojun Chen, Ian H. Sloan, and Robert S. Womersley

SIAM J. Numer. Anal. 48, pp. 2135-2157 (23 pages) | Cited 1 time

Online Publication Date: December 14, 2010

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A set $\mathcal{X}_{N}$ of $N$ points on the unit sphere is a spherical $t$-design if the average value of any polynomial of degree at most $t$ over $\mathcal{X}_{N}$ is equal to the average value of the polynomial over the sphere. This paper considers the characterization and computation of spherical $t$-designs on the unit sphere $\mathbb{S}^2\subset\mathbb{R}^3$ when $N\geq(t+1)^2$, the dimension of the space $\mathbb{P}_t$ of spherical polynomials of degree at most $t$. We show how to construct well conditioned spherical designs with $N\geq(t+1)^2$ points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints. Interval methods are then used to prove the existence of a true spherical $t$-design very close to the calculated points and to provide a guaranteed interval containing the determinant. The resulting spherical designs have good geometrical properties (separation and mesh norm). We discuss the usefulness of the points for both equal weight numerical integration and polynomial interpolation on the sphere and give an example.

Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Harmonic Scattering Problems in Two-Layered Media

Zhiming Chen and Weiying Zheng

SIAM J. Numer. Anal. 48, pp. 2158-2185 (28 pages)

Online Publication Date: December 14, 2010

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In this paper, we propose a uniaxial perfectly matched layer (PML) method for solving the time-harmonic scattering problems in two-layered media. The exterior region of the scatterer is divided into two half spaces by an infinite plane, on two sides of which the wave number takes different values. We surround the computational domain where the scattering field is interested by a PML with the uniaxial medium property. By imposing homogeneous boundary condition on the outer boundary of the PML, we show that the solution of the PML problem converges exponentially to the solution of the original scattering problem in the computational domain as either the PML absorbing coefficient or the thickness of the PML tends to infinity.

Fully Discrete Finite Element Approximations of a Polymer Gel Model

Xiaobing Feng and Yinnian He

SIAM J. Numer. Anal. 48, pp. 2186-2217 (32 pages)

Online Publication Date: December 14, 2010

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This paper proposes and analyzes some fully discrete (multiphysics) finite element methods for a displacement-pressure model which describes swelling dynamics of polymer gels under mechanical constraints. By introducing an “elastic pressure” we first present a reformulation of the original model. We then propose a time-stepping scheme which decouples the PDE system at each time step into two subproblems, one of which is a Stokes-like problem for the displacement vector field (of the solid network of the gel) and the other is a diffusion problem for a “pseudopressure” field (of the solvent of the gel). To make such a multiphysics approach feasible, it is vital to find admissible constraints to resolve the uniqueness issue for the Stokes-like problem and to construct a good boundary condition for the diffusion equation so that it also becomes uniquely solvable. The key to the first difficulty is to discover certain conserved quantities for the PDE solution, and the solution to the second difficulty is to use the Stokes-like problem to generate a boundary condition for the diffusion problem. The time-stepping scheme allows one to use any convergent Stokes solver (and its code) together with any convergent diffusion equation solver (and its code) to solve the polymer gel model. In the paper, the Taylor–Hood mixed finite element method combined with the continuous linear finite element method are chosen as an example to present the ideas and to demonstrate the viability of the proposed multiphysics approach. It is proved that, under a mesh constraint, both the semidiscrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for both semidiscrete and fully discrete methods. Numerical experiments are also presented to show the performance of the proposed approach and methods.

Convergence of the MAC Scheme for the Compressible Stokes Equations

R. Eymard, T. Gallouët, R. Herbin, and J.-C. Latché

SIAM J. Numer. Anal. 48, pp. 2218-2246 (29 pages)

Online Publication Date: December 21, 2010

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We prove in this paper the convergence of the Marker-and-Cell (MAC) scheme for the discretization of the steady state compressible Stokes equations on two- or three-dimensional Cartesian grids. The existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the approximate solutions, up to a subsequence, and in an appropriate sense. We then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations, as well as the equation of state, which is the main difficulty of this study.

The Immersed Interface Technique for Parabolic Problems with Mixed Boundary Conditions

François Bouchon and Gunther H. Peichl

SIAM J. Numer. Anal. 48, pp. 2247-2266 (20 pages)

Online Publication Date: December 21, 2010

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A finite difference scheme is presented for a parabolic problem with mixed boundary conditions. We use an immersed interface technique to discretize the Neumann condition, and we use the Shortley–Weller approximation for the Dirichlet condition. The proof of a discrete maximum principle is given as well as the proof of convergence of the scheme. This convergence is also validated on numerical examples.
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