Numerische Mathematik最新文献

This paper proposes a new numerical method for a fully-coupled, quasi-static thermo-poroelasticity model in a unified enriched Galerkin (EG) method framework. In our method, the mechanics sub-problem is solved using a locking-free EG method, and the flow and heat sub-problems are solved using a locally-conservative EG method. The proposed method offers mass and energy conservation properties with much lower costs than other methods with the same properties, including discontinuous Galerkin methods and mixed finite element methods. The well-posedness and optimal a priori error estimates are carefully derived. Several numerical tests confirm the theoretical optimal convergence rates and the mass and energy conservation properties of the new method.

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala (Comput Methods Appl Mech Eng 320:287–334, 2017) enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the (W^{1,p}) norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection–diffusion problems, all subject to bounds constraints.

We introduce a new discretization based on a polynomial Trefftz-DG method for solving the Stokes equations. Discrete solutions of this method fulfill the Stokes equations pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows us to easily incorporate inhomogeneous right-hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to other (hybrid) discontinuous Galerkin Stokes discretizations and present numerical examples.

Abstract

This paper deals with the equation (-varDelta u+mu u=f) on high-dimensional spaces ({mathbb {R}}^m) , where the right-hand side (f(x)=F(Tx)) is composed of a separable function F with an integrable Fourier transform on a space of a dimension (n>m) and a linear mapping given by a matrix T of full rank and (mu ge 0) is a constant. For example, the right-hand side can explicitly depend on differences (x_i-x_j) of components of x. Following our publication (Yserentant in Numer Math 146:219–238, 2020), we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression (Vert T^tyVert ^2) deviates on the unit sphere (Vert yVert =1) the less from its mean value the higher the dimension m is, a concentration of measure effect. The higher the dimension m, the faster the iteration converges.

We consider a generalization of local density of states which is “windowed” with respect to position and energy, called the windowed local density of states (wLDOS). This definition generalizes the usual LDOS in the sense that the usual LDOS is recovered in the limit where the position window captures individual sites and the energy window is a delta distribution. We prove that the wLDOS is local in the sense that it can be computed up to arbitrarily small error using spatial truncations of the system Hamiltonian. Using this result we prove that the wLDOS is well-defined and computable for infinite systems satisfying some natural assumptions. We finally present numerical computations of the wLDOS at the edge and in the bulk of a “Fibonacci SSH model”, a one-dimensional non-periodic model with topological edge states.

The Fourier-cosine expansion (COS) method is used to price European options numerically in a very efficient way. To apply the COS method, one has to specify two parameters: a truncation range for the density of the log-returns and a number of terms N to approximate the truncated density by a cosine series. How to choose the truncation range is already known. Here, we are able to find an explicit and useful bound for N as well for pricing and for the sensitivities, i.e., the Greeks Delta and Gamma, provided the density of the log-returns is smooth. We further show that the COS method has an exponential order of convergence when the density is smooth and decays exponentially. However, when the density is smooth and has heavy tails, as in the Finite Moment Log Stable model, the COS method does not have exponential order of convergence. Numerical experiments confirm the theoretical results.

We propose and analyze a structure-preserving parametric finite element method (SP-PFEM) for the evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy density (gamma (varvec{n})), where (varvec{n}in mathbb {S}^1) represents the outward unit normal vector. We begin with the anisotropic surface diffusion which possesses two well-known geometric structures—area conservation and energy dissipation—during the evolution of the closed curve. By introducing a novel surface energy matrix (varvec{G}_k(varvec{n})) depending on (gamma (varvec{n})) and the Cahn-Hoffman (varvec{xi })-vector as well as a nonnegative stabilizing function (k(varvec{n})), we obtain a new conservative geometric partial differential equation and its corresponding variational formulation for the anisotropic surface diffusion. Based on the new weak formulation, we propose a full discretization by adopting the parametric finite element method for spatial discretization and a semi-implicit temporal discretization with a proper and clever approximation for the outward normal vector. Under a mild and natural condition on (gamma (varvec{n})), we can prove that the proposed full discretization is structure-preserving, i.e. it preserves the area conservation and energy dissipation at the discretized level, and thus it is unconditionally energy stable. The proposed SP-PFEM is then extended to simulate the evolution of a close curve under other anisotropic geometric flows including anisotropic curvature flow and area-conserved anisotropic curvature flow. Extensive numerical results are reported to demonstrate the efficiency and unconditional energy stability as well as good mesh quality (and thus no need to re-mesh during the evolution) of the proposed SP-PFEM for simulating anisotropic geometric flows.

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem—and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

The method of fundamental solutions has been mainly applied to wave scattering problems in bounded domains and to our knowledge there have not been works addressing density results for general shapes, or addressing the calculation of the complex resonance frequencies that occur in exterior problems. We prove density and convergence of the fundamental solutions approximation in the context of wave scattering problems, with and without a priori knowledge of the frequency, which is of particular importance to detect resonance frequencies for trapping domains. We also present several numerical results that illustrate the good performance of the method in the calculation of complex resonance frequencies for trapping and non trapping domains in 2D and 3D.

In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given time step does not necessarily use all the parts of the split operator. This is in contrast to deterministic splitting schemes which always use every part at least once, and often several times. As a result, the computational cost can be significantly decreased in comparison to such methods. We rigorously define a randomized operator splitting scheme in an abstract setting and provide an error analysis where we prove that the temporal convergence order of the scheme is at least 1/2. We illustrate the theory by numerical experiments on both linear and quasilinear diffusion problems, using a randomized domain decomposition approach. We conclude that choosing the randomization in certain ways may improve the order to 1. This is as accurate as applying e.g. backward (implicit) Euler to the full problem, without splitting.