Prior Error Estimates Research Articles

Analysis of a double Timoshenko beam model

In this work we consider a double beam system modeled in the theory of Timoshenko. An existence and uniqueness result is achieved by using the standard theory of linear semigroup. The exponential stability is also proved. Then, fully discrete approximations are introduced and a prior error estimates are shown. Finally, some numerical simulations are presented.

Shape optimization of Stokes flows by a penalty method

Shape optimization in incompressible Stokes flows is considered based on the penalty method for the divergence free constraint at continuous level. Shape sensitivity analysis is performed and numerical algorithms are presented. An iterative penalty method with reliable accuracy is used for solving numerically the penalized state and possible adjoint. The iterative penalty method as an inner solver is more efficient than the standard mixed finite element method in 2D. Asymptotic convergence analysis and a prior error estimates for finite element discretizations of both state and adjoint are provided. Numerical results are presented to show the effectiveness of the numerical optimization algorithms.

α-Robust H1-norm analysis of a finite element method for the superdiffusion equation with weak singularity solutions

In this work, the superdiffusion equation with a Caputo derivative of order α∈(1,2) is considered. Some priori bounds on certain derivatives of the solution show that the solution exhibits a weak singularity at the initial time t=0. To resolve this initial singularity, we rewrite the superdiffusion equation as a coupled system by introducing a intermediate variable p:=Dtα/2(u−tu1), and adopt the L1 scheme and Alikhanov scheme on graded meshes in temporal direction. In spatial direction, the conforming finite element method is used. Furthermore, we derive the H1-norm stability result. It is worth noting that some priori bounds on certain derivatives of p are obtained, on basis of which, we derive an α-robust prior error estimate with optimal H1-norm convergence order. Finally, we provide the numerical experiment to further verify our theoretical analysis.

Numerical Solution of the Heat Transfer Equation Coupled with the Darcy Flow Using the Finite Element Method

The finite element approach was utilized in this study to solve numerically the two-dimensional time-dependent heat transfer equation coupled with the Darcy flow. The Picard-Lindelöf Theorem was used to prove the existence and uniqueness of the solution. The prior and posterior error estimates are then derived for the numerical scheme. Numerical examples were provided to show the effectiveness of the theoretical results. The essential code development in this study was done using MATLAB computer simulation.

Conforming and nonconforming VEMs for the fourth-order reaction–subdiffusion equation: a unified framework

AbstractWe establish a unified framework to study the conforming and nonconforming virtual element methods (VEMs) for a class of time dependent fourth-order reaction–subdiffusion equations with the Caputo derivative. To resolve the intrinsic initial singularity we adopt the nonuniform Alikhanov formula in the temporal direction. In the spatial direction three types of VEMs, including conforming virtual element, $C^0$ nonconforming virtual element and fully nonconforming Morley-type virtual element, are constructed and analysed. In order to obtain the desired convergence results, the classical Ritz projection operator for the conforming virtual element space and two types of new Ritz projection operators for the nonconforming virtual element spaces are defined, respectively, and the projection errors are proved to be optimal. In the unified framework we derive a prior error estimate with optimal convergence order for the constructed fully discrete schemes. To reduce the computational cost and storage requirements, the sum-of-exponentials (SOE) technique combined with conforming and nonconforming VEMs (SOE-VEMs) are built. Finally, we present some numerical experiments to confirm the theoretical correctness and the effectiveness of the discrete schemes. The results in this work are fundamental and can be extended into more relevant models.

Interior penalty discontinuous Galerkin technique for solving generalized Sobolev equation

This paper proposes a discontinuous Galerkin method to solve the generalized Sobolev equation. In this numerical procedure, the temporal variable has been discretized by the Crank–Nicolson idea to get a time-discrete scheme with the second-order accuracy. Then, in the second stage the spatial variable has been discretized by the discontinuous Galerkin finite element method. A prior error estimate has been proposed for the semi-discrete scheme based on the spatial discretization. By applying the Crank–Nicolson idea a full-discrete scheme is driven. Furthermore, an error estimate has been proved to get the convergence order of the developed scheme. Finally, some numerical examples have been presented to show the efficiency and theoretical results of the new numerical procedure.

A continuous Galerkin method for pseudo-hyperbolic equations with variable coefficients

This paper, we study a space-time continuous Galerkin (STCG) method with mesh change for two-dimensional (2-D) pseudo-hyperbolic equations with variable coefficients, which gives the same treatment for time and space discretizations, namely both time and space variables are discretized via finite element (FE) method. In addition, it allows the change of time step and space mesh which are necessary for adaptive algorithms. We prove the existence and uniqueness of approximation solution and give a prior error estimate without any conditions attached to the space-time grid. Finally, a numerical example is given to confirm the feasibility to the scheme constructed here.

Local discontinuous Galerkin methods for parabolic interface problems with homogeneous and non-homogeneous jump conditions

In this paper, we present the local discontinuous Galerkin (LDG) methods for solving second order parabolic interface problems in two-dimensional convex polygonal domains with homogeneous and non-homogeneous jump conditions, respectively. We start analyzing from the homogeneous problems. After giving the semi discrete LDG method, we derive its stability and a prior error estimate. Then, after a series of derivation, we obtain the LDG method for the non-homogeneous problems, which takes the same formulation as the homogeneous case with a special choice of numerical fluxes that incorporate the jump conditions across interface. So its analysis can be easily obtained within the framework developed for the homogeneous case. It is an advantage of the LDG method that it provides a natural framework to enforce the discontinuities in both the potential and the flux weakly in the discrete formulation provided that the triangulation of the domain is fitted to the interface. Numerical experiments are conducted to confirm our theoretical analysis with the help of the explicit temporal discretization and show that the LDG methods have the optimal and suboptimal convergence rate in the L2 norms for the potential and the flux, respectively. In addition, they also indicate that the LDG methods possess the same convergence properties in the L∞ norms. What is more, to avoid the strict time-step restriction of explicit schemes, the implicit integration factor (IIF) method is employed, which not only relaxes the time step to Δt=O(hmin) but also remains the important property of LDG method that the computation can proceed element by element and avoid solving a global system of algebraic equations as the standard implicit schemes do. We provide numerical examples to illustrate that the numerical scheme combined LDG method with IIF method indeed reduces the computational cost greatly.

A coupled electro-thermal Discontinuous Galerkin method

This paper presents a Discontinuous Galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro-thermal problems.In this paper we discuss the fundamental equations for the transport of electricity and heat, in terms of macroscopic variables such as temperature and electric potential. A fully coupled nonlinear weak formulation for electro-thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients. The weak form can thus be formulated as a Discontinuous Galerkin method.The existence and uniqueness of the weak form solution are proved. The numerical properties of the nonlinear elliptic problems i.e., consistency and stability, are demonstrated under specific conditions, i.e. use of high enough stabilization parameter and at least quadratic polynomial approximations. Moreover the prior error estimates in the H1-norm and in the L2-norm are shown to be optimal in the mesh size with the polynomial approximation degree.

SU‐E‐T‐769: T‐Test Based Prior Error Estimate and Stopping Criterion for Monte Carlo Dose Calculation in Proton Therapy

Purpose:The Monte Carlo (MC) method is a gold standard for dose calculation in radiotherapy. However, it is not a priori clear how many particles need to be simulated to achieve a given dose accuracy. Prior error estimate and stopping criterion are not well established for MC. This work aims to fill this gap.Methods:Due to the statistical nature of MC, our approach is based on one‐sample t‐test. We design the prior error estimate method based on the t‐test, and then use this t‐test based error estimate for developing a simulation stopping criterion. The three major components are as follows.First, the source particles are randomized in energy, space and angle, so that the dose deposition from a particle to the voxel is independent and identically distributed (i.i.d.).Second, a sample under consideration in the t‐test is the mean value of dose deposition to the voxel by sufficiently large number of source particles. Then according to central limit theorem, the sample as the mean value of i.i.d. variables is normally distributed with the expectation equal to the true deposited dose.Third, the t‐test is performed with the null hypothesis that the difference between sample expectation (the same as true deposited dose) and on‐the‐fly calculated mean sample dose from MC is larger than a given error threshold, in addition to which users have the freedom to specify confidence probability and region of interest in the t‐test based stopping criterion.Results:The method is validated for proton dose calculation. The difference between the MC Result based on the t‐test prior error estimate and the statistical Result by repeating numerous MC simulations is within 1%.Conclusion:The t‐test based prior error estimate and stopping criterion are developed for MC and validated for proton dose calculation.Xiang Hong and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang Talent Program (#14PJ1404500).

Numerical analysis of a pseudo-compact C-N conservative scheme for the Rosenau-KdV equation coupling with the Rosenau-RLW equation

In this paper, numerical solutions for the Rosenau-KdV equation coupling with the Rosenau-RLW equation are considered and a new C-N pseudo-compact conservative numerical scheme, which preserves the original conservative properties is designed. The proposed scheme is based on a finite difference method. The existence of the difference solutions has been shown by the Brouwer fixed point theorem. Unconditional stability, second-order convergence, and a prior error estimate of the scheme are proved by the discrete energy method. Numerical examples have been given to verify the theoretical results.

Estimates of Analysis and Forecast Error Variances Derived from the Adjoint of 4D-Var

Abstract A method is presented in which the adjoint of a four-dimensional variational data assimilation system (4D-Var) was used to compute the expected analysis and forecast error variances of linear functions of the ocean state vector. The power and utility of the approach are demonstrated using the Regional Ocean Modeling System configured for the California Current system. Linear functions of the ocean state vector were considered in the form of indices that characterize various aspects of the coastal upwelling circulation. It was found that for configurations of 4D-Var typically used in ocean models, reliable estimates of the expected analysis error variances can be obtained both for variables that are observed and unobserved. In addition, the contribution of uncertainties in the model control variables to the forecast error variance was also quantified. One particularly powerful and illuminating aspect of the adjoint 4D-Var approach to the forecast problem is that the contribution of individual observations to the predictability of the circulation can be readily computed. An important finding of the work presented here is that despite the plethora of available satellite observations, the relatively modest fraction of in situ subsurface observations sometimes exerts a significant influence on the predictability of the coastal ocean. Independent checks of the analysis and forecast error variances are also presented, which provide a direct test of the hypotheses that underpin the prior error and observation error estimates used during 4D-Var.

Local projection stabilized finite element method for Navier-Stokes equations

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier-Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well.

A two-grid method for finite volume element approximations of second-order nonlinear hyperbolic equations

The two-grid method is studied for solving a two-dimensional second-order nonlinear hyperbolic equation using finite volume element method. The method is based on two different finite element spaces defined on one coarse grid with grid size H and one fine grid with grid size h , respectively. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. A prior error estimate in the H 1 -norm is proved to be O ( h + H 3 | ln H | ) for the two-grid semidiscrete finite volume element method. With these proposed techniques, solving such a large class of second-order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation. Finally, a numerical example is presented to validate the usefulness and efficiency of the method.

An error analysis of a finite element method for a system of nonlinear advection–diffusion–reaction equations

In this paper we present an analysis of a finite element method for solving a system of nonlinear advection–diffusion–reaction equations. We prove the existence and uniqueness of the numerical solution, and obtain a prior error estimates optimal in H 1 norm. Numerical computations are carried out to test the order of accuracy in the error estimates. Blend function interpolations are applied in the calculation of numerical integrations over elements with a curved boundary.

An upwind finite volume element method based on quadrilateral meshes for nonlinear convection‐diffusion problems

AbstractConsidering an upwind finite volume element method based on convex quadrilateral meshes for computing nonlinear convection‐diffusion problems, some techniques, such as calculus of variations, commutating operator, and the theory of prior error estimates and techniques, are adopted. Discrete maximum principle and optimal‐order error estimates in H1 norm for fully discrete method are derived to determine the errors in the approximate solution. Thus, the well‐known problem [(Li et al., Generalized difference methods for differential equations: numerical analysis of finite volume methods, Marcel Dekker, New York, 2000), p 365.] has been solved. Some numerical experiments show that the method is a very effective engineering computing method for avoiding numerical dispersion and nonphysical oscillations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009

New techniques in designing finite-difference domain decomposition algorithm for the heat equation

This paper presents some new techniques in designing finite-difference domain decomposition algorithm for the heat equation. The basic procedure is to define the finite-difference schemes at the interface grid points with smaller time step Δ t = Δt m ( m is a positive integer) by Saul'yev asymmetric schemes. The algorithm can increase the stability bounds of the classical explicit method by 2 m times, and the prior error estimates for the numerical solutions are obtained for some algorithms when m = 2 or m = 3. Numerical experiments on stability and accuracy are also presented.

On a finite element method for unsteady compressible viscous flows

AbstractIn this article we present an analysis of a finite element method for solving two‐dimensional unsteady compressible Navier‐Stokes equations. Under the time‐stepping size restriction Δt ≤ Ch, we prove the existence and uniqueness of the numerical solution and obtain an a prior error estimate uniform in time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 152–166, 2003

III: OCEAN CIRCULATION: Global Ocean Data Assimilation and Geoid Measurements

Parts of geodesy and physical oceanography are about to mature into a single modeling problem involving the simultaneous estimation of the marine geoid and the general circulation. Both fields will benefit. To this end, we present an ocean state estimation (data assimilation) framework which is designed to obtain a dynamically consistent picture of the changing ocean circulation by combining global ocean data sets of arbitrary type with a general circulation model (GCM). The impact of geoid measurements on such estimates of the ocean circulation are numerous. For the mean circulation, a precise geoid describes the reference frame for dynamical signals in altimetric sea surface height observations. For the time-varying ocean signal, changing geoid information might be a valuable new information about correcting the changing flow field on time scales from a few month to a year, but the quantitative utility of such information has not yet been demonstrated. For a consistent estimate, some knowledge of the prior error covariances of all data fields is required. The final result must be consistent with prior error estimates for the data. State estimation is thus one of the few quantitative consistency checks for new geoid measurements anticipated from forthcoming space missions. Practical quantitative methods will yield a best possible estimate of the dynamical sea surface which, when combined with satellite altimetric surfaces, will produce a best-estimate marine geoid. The anticipated accuracy and precision of such estimates raises some novel modeling error issues which have not conventionally been of concern (the Boussinesq approximation, self-attraction and loading). Model skill at very high frequencies is a major concern because of the need to de-alias the data obtained by the inevitable oceanic temporal undersampling dictated by realistic satellite orbit configurations.

Generalized Inversion of Tropical Atmosphere–Ocean (TAO) Data and a Coupled Model of the Tropical Pacific. Part II: The 1995–96 La Niña and 1997–98 El Niño

Abstract The investigation of the consequences of trying to blend tropical Pacific observations from the Tropical Atmosphere–Ocean (TAO) array into the dynamical framework of an intermediate coupled ocean–atmosphere model is continued. In a previous study it was found that the model dynamics, the prior estimates of uncertainty in the observations, and the estimates of the errors in the dynamical equations of the model could not be reconciled with data from the 1994–95 period. The error estimates and the data forced the rejection of the model physics as being unacceptably in error. In this work, data from two periods (1995–96 and 1997–98) were used when the tropical Pacific was in states very different from the previous study. The consequences of increasing the prior error estimates were explored in an effort to find out if it is possible at least to use the intermediate model physics to assist in mapping the observations into fields in a statistically consistent way. It was found that such a result is pos...