Finite Element Spaces Research Articles

A Partition of unity finite element method for valuation American option under Black-Scholes model

Abstract In this paper, we present an intelligent combination of partition of unity (PU) and finite element (FE) methods for valuing American option pricing problems governed by the Black-Scholes (BS) model. The model is based on a partial differential equation (PDE) from which one can deduce the Black-Scholes formula, which gives a theoretical estimated value of options using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration and expected volatility. Although the finite element method (FEM) seems to be an alternative tool for pricing options with a few applications reported in the literature, this combination called the Partition of Unity Finite Element Method (PUFEM) appears to offer many of the desired properties. The main advantage of the proposed approach is its ability to locally refine the solution by adapting an incorporated specific class of enrichment in the finite element space instead of generating a new fine mesh for the problem under study. Numerical computations are carried out to show a huge reduction in the number of degrees of freedom required to achieve a fixed accuracy which confirms that the PUFE method used is very efficient and gives better accuracy than the conventional FE method.

Duality in Finite Element Exterior Calculus and Hodge Duality on the Sphere

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension n, arbitrary polynomial degree r, and arbitrary differential form degree k. The study of finite element exterior calculus began with the $${\mathcal {P}}_r\varLambda ^k$$ P r Λ k and $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between $${\mathcal {P}}_r\varLambda ^k$$ P r Λ k and $$\mathring{{\mathcal {P}}}_{r+k+1}^-\varLambda ^{n-k}$$ P ˚ r + k + 1 - Λ n - k and between $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k and $$\mathring{{\mathcal {P}}}_{r+k}\varLambda ^{n-k}$$ P ˚ r + k Λ n - k . In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard n-sphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the $${\mathcal {P}}_r\varLambda ^k$$ P r Λ k , $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k , $$\mathring{{\mathcal {P}}}_r\varLambda ^k$$ P ˚ r Λ k and $$\mathring{{\mathcal {P}}}_r^-\varLambda ^k$$ P ˚ r - Λ k spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between $${\mathcal {P}}_r \varLambda ^k$$ P r Λ k and $$\mathring{{\mathcal {P}}}_{r+k+1}^-\varLambda ^{n-k}$$ P ˚ r + k + 1 - Λ n - k and between $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k and $$\mathring{{\mathcal {P}}}_{r+k} \varLambda ^{n-k}$$ P ˚ r + k Λ n - k . These isomorphisms can be implemented via a simple computation, which we illustrate with examples.

Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity

AbstractThis paper proposes and analyzes a posteriori error estimator based on stress equilibration for linear elasticity with emphasis on the behavior for (nearly) incompressible materials. It is based on an H(div)‐conforming, weakly symmetric stress reconstruction from the displacement‐pressure approximation computed with a stable finite element pair. Our focus is on the Taylor‐Hood combination of continuous finite element spaces of polynomial degrees k + 1 and k for the displacement and the pressure, respectively. This weak symmetry allows us to prove that the resulting error estimator constitutes a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. It does not involve global constants like those from Korn's in equality which may become very large depending on the location and type of the boundary conditions. Local efficiency, also uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator. Numerical results for the popular Cook's membrane test problem confirm the theoretical predictions.

Reliability and Efficiency of DWR-Type A Posteriori Error Estimates with Smart Sensitivity Weight Recovering

Abstract We derive efficient and reliable goal-oriented error estimations, and devise adaptive mesh procedures for the finite element method that are based on the localization of a posteriori estimates. In our previous work [B. Endtmayer, U. Langer and T. Wick, Two-side a posteriori error estimates for the dual-weighted residual method, SIAM J. Sci. Comput. 42 2020, 1, A371–A394], we showed efficiency and reliability for error estimators based on enriched finite element spaces. However, the solution of problems on an enriched finite element space is expensive. In the literature, it is well known that one can use some higher-order interpolation to overcome this bottleneck. Using a saturation assumption, we extend the proofs of efficiency and reliability to such higher-order interpolations. The results can be used to create a new family of algorithms, where one of them is tested on three numerical examples (Poisson problem, p-Laplace equation, Navier–Stokes benchmark), and is compared to our previous algorithm.

An $$\alpha $$-Robust Semidiscrete Finite Element Method for a Fokker–Planck Initial-Boundary Value Problem with Variable-Order Fractional Time Derivative

A time-fractional initial-boundary value problem of Fokker–Planck type is considered on the space-time domain $$\Omega \times [0,T]$$ , where $$\Omega $$ is an open bounded domain in $$\mathbb {R}^d$$ for some $$d\ge 1$$ , and the order $$\alpha (x)$$ of the Riemann-Liouville fractional derivative may vary in space with $$1/2< \alpha (x) < 1$$ for all x. Such problems appear naturally in the formulation of certain continuous-time random walk models. Uniqueness of any solution u of the problem is proved under reasonable hypotheses. A semidiscrete numerical method, using finite elements in space to yield a solution $$u_h(t)$$ , is constructed. Error estimates for $$\Vert (u - u_h)(t)\Vert _{L^2(\Omega )}$$ and $$\int _0^t \left| \partial _t^{1-\alpha } (u-u_h)(s)\right| _1^2 \,ds$$ are proved for each $$t\in [0,T]$$ under the assumptions that the following quantities are finite: $$\Vert u(\cdot , 0)\Vert _{H^2(\Omega )}, |u(\cdot , t)|_{H^1(\Omega )}$$ for each t, and $$\int _0^t [\Vert u(\cdot , t)\Vert _{H^2(\Omega )}^2 + |\partial _t^{1-\alpha }u|_{H^2(\Omega )}^2]$$ , where u(x, t) is the unknown solution. Furthermore, these error estimates are $$\alpha $$ -robust: they do not fail when $$\alpha \rightarrow 1$$ , the classical Fokker–Planck problem. Sharper results are obtained for the special case where the drift term of the problem is not present (which is of interest in certain applications).

Convergence and Superconvergence of Fully Discrete Finite Element for Time Fractional Optimal Control Problems

In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and L1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.

A Structure-Preserving Parametric Finite Element Method for Surface Diffusion

We propose a structure-preserving parametric finite element method (SP-PFEM) for discretizing the surface diffusion of a closed curve in two dimensions (2D) or surface in three dimensions (3D). Here the "structure-preserving" refers to preserving the two fundamental geometric structures of the surface diffusion flow: (i) the conservation of the area/volume enclosed by the closed curve/surface, and (ii) the decrease of the perimeter/total surface area of the curve/surface. For simplicity of notations, we begin with the surface diffusion of a closed curve in 2D and present a weak (variational) formulation of the governing equation. Then we discretize the variational formulation by using the backward Euler method in time and piecewise linear parametric finite elements in space, with a proper approximation of the unit normal vector by using the information of the curves at the current and next time step. The constructed numerical method is shown to preserve the two geometric structures and also enjoys the good property of asymptotic equal mesh distribution. The proposed SP-PFEM is "weakly" implicit (or almost semi-implicit) and the nonlinear system at each time step can be solved very efficiently and accurately by the Newton's iterative method. The SP-PFEM is then extended to discretize the surface diffusion of a closed surface in 3D. Extensive numerical results, including convergence tests, structure-preserving property and asymptotic equal mesh distribution, are reported to demonstrate the accuracy and efficiency of the proposed SP-PFEM for simulating surface diffusion in 2D and 3D.

A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system

&lt;p style='text-indent:20px;'&gt;In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with an ODE that has to hold true in almost all points in space. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the discretization of the control we employ the same discretization technique which turns out to be equivalent to a variational discretization approach. We provide error estimates of optimal order both for the discretization of the state equation as well as for the optimal control. Numerical experiments are added to illustrate the proven rates of convergence.&lt;/p&gt;

The Aggregated Unfitted Finite Element Method on Parallel Tree-Based Adaptive Meshes

In this work, we present an adaptive unfitted finite element scheme that combines the aggregated finite element method with parallel adaptive mesh refinement. We introduce a novel scalable distributed-memory implementation of the resulting scheme on locally adapted Cartesian forest-of-trees meshes. We propose a two-step algorithm to construct the finite element space at hand by means of a discrete extension operator that carefully mixes aggregation constraints of problematic degrees of freedom, which get rid of the small cut cell problem, and standard hanging degree of freedom constraints, which ensure trace continuity on nonconforming meshes. Following this approach, we derive a finite element space that can be expressed as the original one plus well-defined linear constraints. Moreover, it requires minimum parallelization effort, using standard functionality available in existing large-scale finite element codes. Numerical experiments demonstrate its optimal mesh adaptation capability, robustness to cut location, and parallel efficiency, on classical Poisson hp-adaptivity benchmarks. Our work opens the path to functional and geometrical error-driven dynamic mesh adaptation with the aggregated finite element method in large-scale realistic scenarios. Likewise, it can offer guidance for bridging other scalable unfitted methods and parallel adaptive mesh refinement.

An adjoint-based a posteriori analysis of numerical approximation of Richards equation

&lt;p style='text-indent:20px;'&gt;This paper formulates a general framework for a space-time finite element method for solving Richards Equation in one spatial dimension, where the spatial variable is discretized using the linear finite volume element and the temporal variable is discretized using a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in temporal variable to the variational equation and appropriate "variational crime" in the form of numerical integrations for calculating integrations in the formulation. Once this is in place, adjoint-based error estimators for the approximate solution from the scheme is derived. The adjoint problem is obtained from an appropriate linearization of the nonlinear system. Numerical examples are presented to illustrate performance of the methods and the error estimators.&lt;/p&gt;

Geometric Variational Finite Element Discretizations for Fluids

Abstract We present an overview of finite element variational integrators for compressible and incompressible fluids with variable density. The numerical schemes are derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart-Thomas finite element space. We illustrate the conservation properties of the scheme with the Rayleigh-Taylor instability test.

On the Sobolev and $L^p$-Stability of the $L^2$-Projection

We show stability of the $L^2$-projection onto Lagrange finite element spaces with respect to (weighted) $L^p$ and $W^{1,p}$-norms for any polynomial degree and for any space dimension under suitable conditions on the mesh grading. This includes $W^{1,2}$-stability in two space dimensions for any polynomial degree and meshes generated by newest vertex bisection. Under realistic but conjectured assumptions on the mesh grading in three dimensions we show $W^{1,2}$-stability for all polynomial degrees. We also propose a modified bisection strategy that leads to better $W^{1,p}$-stability. Moreover, we investigate the stability of the $L^2$-projection onto Crouzeix-Raviart elements.

A Hybridized High-Order Method for Unique Continuation Subject to the Helmholtz Equation

We design and analyze an arbitrary-order hybridized discontinuous Galerkin method to approximate the unique continuation problem subject to the Helmholtz equation. The method is analyzed using conditional stability estimates for the continuous problem, leading to error estimates in norms over interior subdomains of the computational domain. The convergence order reflects the Holder continuity of the conditional stability estimates and the approximation properties of the finite element space for sufficiently smooth solutions. Under a certain convexity condition, the constant in the estimates is independent of the frequency. Moreover, certain weighted averages of the error are shown to converge independently of the stability properties of the continuous problem. Numerical examples illustrate the performances of the method with respect to the degree of ill-posedness of the problem, increasing polynomial order and perturbations in the data.

Mixed methods for degenerate elliptic problems and application to fractional Laplacian

We analyze the approximation by mixed finite element methods of solutions of equations of the form −div (a∇u) = g, where the coefficient a = a(x) can degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the coefficient a belongs to the Muckenhoupt class A2. The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart–Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes. For the lowest order case we show that the regularity assumption can be removed and prove anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.

Additive Schwarz preconditioners for a localized orthogonal decomposition method

We investigate a variant of the localized orthogonal decomposition method (Henning and Peterseim, [Multiscale Model. Simul., 11 (2013), pp. 1149–1175] and Målqvist and Peterseim, [Math. Comp., 83 (2014), pp. 2583–2603]) for elliptic problems with rough coefficients. The construction of the basis of the multiscale finite element space is based on domain decomposition techniques, which is motivated by the recent work of Kornhuber, Peterseim, and Yserentant [Math. Comp., 87 (2018), pp. 2765–2774]. We also design and analyze additive Schwarz domain decomposition preconditioners for the resulting discrete problems.

Divergence-free Scott--Vogelius Elements on Curved Domains

We construct and analyze an isoparametric finite element pair for the Stokes problem in two dimensions. The pair is defined by mapping the Scott--Vogelius finite element space via a Piola transform...

An [formula omitted]-robust finite element method for a multi-term time-fractional diffusion problem

A time-fractional initial–boundary problem is considered on a bounded spatial domain Ω⊂Rd, where d∈{1,2,3} and Ω is convex or smooth. The differential equation is ∑i=1lqiDtαiu(x,t)−Δu(x,t)=f, where each Dtαi is a Caputo derivative with 0<αl<⋯<α1<1 and the qi are positive constants. A new error analysis is given for the numerical method (L1 scheme in time, finite elements in space) of Huang and Stynes (2020). The error bounds that are derived here remain valid (i.e., are “α-robust”) if α1→1−, unlike the bounds in Huang and Stynes (2020), which blow up as α1→1−. Thus these new error bounds are a more natural and desirable result, since the unknown solution is well behaved as α1→1−.

Impact of power law fluid and magnetic field on double diffusive mixed convection in staggered porous cavity considering Dufour and Soret effects

This study deals the influence of power law fluid and inclined magnetic field on a porous medium in staggered cavity. The effect of Soret and Dufour parameters are also given attention. A two dimensional system of partial differential equations has been discretized by employing Galerkin finite element method. A finite element method involving the cubic polynomials (ℙ3) has been implemented to compute for velocity, temperature and concentration fields while the pressure is approximated by quadratic (ℙ2) finite element space of functions. The system of discretized equations is simplified using the adaptive Newton's method. Simulations are performed for various ranges of pertinent parameters such as power law index (between 0.6 and 1.8), Hartmann number (between 0 and 100), Lewis number (between 1 and 10), Dufour/Soret numbers (between −3 and 3), Darcy number (between 10−5 and 10−2), magnetic field inclination (between 0∘ and 90∘) and buoyancy ratio (between 0 and 10). It is inferred that the retarding effect of the rising Lorentz force is pronounced on fluid flow. The convective heat transfer is enhanced with the change of Dufour number from negative to positive. The suppression on convection at the inclination angle γ = 90∘ is much more than γ = 0∘. The increment in Dufour and Soret numbers has an improving influence on both the average Nusselt and Sherwood numbers, and thus the convective heat and mass transfer at buoyancy ratio N = 10.

P 1-nonconforming divergence-free finite element method on square meshes for Stokes equations

Abstract Recently, the P 1-nonconforming finite element space over square meshes has been proved stable to solve Stokes equations with the piecewise constant space for velocity and pressure, respectively. In this paper, we will introduce its locally divergence-free subspace to solve the elliptic problem for the velocity only decoupled from the Stokes equation. The concerning system of linear equations is much smaller compared to the Stokes equations. Furthermore, it is split into two smaller ones. After solving the velocity first, the pressure in the Stokes problem can be obtained by an explicit method very rapidly.

Modeling effectiveness and identification of multi-scale objects in farmland soils with improved Yee-FDTD methods

Finite-Difference Time-Domain (FDTD) is the most popular time-domain approach in computational electromagnetics. Due to the Courant-Friedrich-Levy (CFL) condition and the perfect match layer (PML) boundary precision, FDTD cannot simulate soil medium whose surface is connected by multiple straight lines or curves (multi-scale) accurately and efficiently, which greatly limits the application of FDTD method to simulate buried objects in soils. Firstly, this study proposed the absorption boundary and adopted two typical perfect matching layers (UPML and CPML) to compare their absorption effects, and then using the three forms of improved Yee-FDTD algorithm, alternating-direction implicit (ADI-FDTD), unconditionally stable (US-FDTD) and hybrid implicit explicit finite time-domain (HIE-FDTD) to divide and contrast the boundary model effects. It showed that the HIE-FDTD was suitable for inversion of multi-scale structure object modeling, while ADI-FDTD and US-FDTD were ideal for single-boundary objects in both uniaxial perfectly matched layer (UMPL) and convolution perfectly matched layer (CPML) finite element space. After that, all the models were tested by computer performance for their simulated efficiency. When simulating single boundary objects, UPML-US-FDTD and ADI-FDTD could achieve the ideal results, and in the boundary inversion of multi-scale objects, HIE-FDTD modeling results and efficiency were the best. Test modeling speeds of CPML-HIE-FDTD were compared with three kinds of waveform sources, Ricker, Blackman-Harris and Gaussian. Finally, under the computer condition in which the CPU was i5-8250, the HIE-FDTD model still had better performance than the traditional Yee-FDTD forward modeling algorithm. For modeling multi-scale objects in farmland soils, the methods used CPML combined with the HIE-FDTD were the most efficient and accurate ways. This study can solve the problem that the traditional FDTD algorithm cannot construct non-mesh objects by utilizing the diversity characteristics of Yee cell elements. Keywords: Yee-FDTD, multi-scale objects, modeling effectiveness, Ground Penetrating Radar, farmland soils DOI: 10.25165/j.ijabe.20201306.5443 Citation: Li Y H, Zhao Z X, Qiu Z, Luo Y F, Zhu Y C. Modeling effectiveness and identification of multi-scale objects in farmland soils with improved Yee-FDTD methods. Int J Agric & Biol Eng, 2020; 13(6): 150–158.