Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises.The noise term is approximated through the spectral projection of the covariance operator,which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises,the well-posedness of the SPDE is established under certain covariance operator-dependent conditions.These SPDEs with projected noises are then numerically approximated with the finite element method.A general error estimate framework is established for the finite element approximations.Based on this framework,optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained.It is shown that with the proposed approach,convergence order of white noise driven SPDEs is improved by half for one-dimensional problems,and by an infinitesimal factor for higher-dimensional problems.

ADAPTIVE FINITE ELEMENT APPROXIMATION FOR A CLASS OF PARAMETER ESTIMATION PROBLEMS

In this paper, we study adaptive finite element discretisation schemes for a class of parameter estimation problem. We propose to efficient algorithms for the estimation problem use adaptive multi-meshes in developing We derive equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.

Finite Element Approximation of Semilinear Parabolic Optimal Control Problems

In this paper,the finite element approximation of a class of semilinear parabolic optimal control problems with pointwise control constraint is studied.We discretize the state and co-state variables by piecewise linear continuous functions,and the control variable is approximated by piecewise constant functions or piecewise linear discontinuous functions.Some a priori error estimates are derived for both the control and state approximations.The convergence orders are also obtained.

FINITE ELEMENT APPROXIMATION FOR A CLASS OF PARAMETER ESTIMATION PROBLEMS

This paper investigates the finite element approximation of a class of parameter estimation problems which is the form of performance as the optimal control problems governed by bilinear parabolic equations,where the state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions.The authors derive some a priori error estimates for both the control and state approximations.Finally,the numerical experiments verify the theoretical results.

FINITE ELEMENT APPROXIMATION OF EIGENVALUE PROBLEM FOR A COUPLED VIBRATION BETWEEN ACOUSTIC FIELD AND PLATE

We formulate a coupled vibration between plate and acoustic field in mathematically rigorous fashion. It leads to a non-standard eigenvalue problem. A finite element approximation is considered in an abstract way, and the approximate eigenvalue problem is written in an operator form by means of some Ritz projections. The order of convergence is proved based on the result of Babugka and Osborn. Some numerical example is shown for the problem for which the exact analytical solutions are calculated. The results shows that the convergence order is consistent with the one by the numerical analysis.

ON THE ERROR ESTIMATE OF LINEAR FINITE ELEMENT APPROXIMATION TO THE ELASTIC CONTACT PROBLEM WITH CURVED CONTACT BOUNDARY

In this paper, the linear finite element approximation to the elastic contact problem with curved contact boundary is considered. The error bound O(h[sup ?]) is obtained with requirements of two times continuously differentiable for contact boundary and the usual regular triangulation, while I.Hlavacek et. al. Obtained the error bound O(h[sup ?]) with requirements of three times continuously differentiable for contact boundary and extra regularities of triangulation (c.f. [2]). [ABSTRACT FROM AUTHOR]

Finite Element Convergence for State-Based Peridynamic Fracture Models

We establish the a priori convergence rate for finite element approximations of a class of nonlocal nonlinear fracture models.We consider state-based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of the nonlocal interaction.The pairwise interactions between points are mediated by a bond potential of multi-well type while multi-point interactions are associated with the volume change mediated by a hydrostatic strain potential.The hydrostatic potential can either be a quadratic function,delivering a linear force–strain relation,or a multi-well type that can be associated with the material degradation and cavitation.We first show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space H2.We show that the finite element approximations converge to the H2 solutions uniformly as measured in the mean square norm.For linear continuous fi nite elements,the convergence rate is shown to be Ct Δt+Csh2/ε2,where𝜖is the size of the horizon,his the mesh size,and Δt is the size of the time step.The constants Ct and Cs are independent of Δt and h and may depend on ε through the norm of the exact solution.We demonstrate the stability of the semi-discrete approximation.The stability of the fully discrete approximation is shown for the linearized peridynamic force.We present numerical simulations with the dynamic crack propagation that support the theoretical convergence rate.

Approximation of thermoelasticity contact problem with nonmonotone friction

The paper presents the formulation and approximation of a static thermoelasticity problem that describes bilateral frictional contact between a deformable body and a rigid foundation. The friction is in the form of a nonmonotone and multivalued law. The coupling effect of the problem is neglected. Therefore, the thermic part of the problem is considered independently on the elasticity problem. For the displacement vector, we formulate one substationary problem for a non-convex, locally Lipschitz continuous functional representing the total potential energy of the body. All problems formulated in the paper are approximated with the finite element method.

SUPERCONVERGENCE ANALYSIS OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS OF THE STATIONARY B(?)NARD TYPE

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in L^∞-norm and optimal error estimates in L^2-norm.

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.

Recovery Type A Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints.The time discretization is based on the backward Euler method.The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions.We derive the superconvergence properties of finite element solutions.By using the superconvergence results,we obtain recovery type a posteriori error estimates.Some numerical examples are presented to verify the theoretical results.

LEAST-SQUARES MIXED FINITE ELEMENT METHOD FOR SADDLE-POINT PROBLEM

In this paper, a least-squares mixed finite element method for the solution of the primal saddle-point problem is developed. It is proved that the approximate problem is consistent ellipticity in the conforming finite element spaces with only the discrete BB-condition needed for a smaller auxiliary problem. The abstract error estimate is derived. [ABSTRACT FROM AUTHOR]

A FINITE ELEMENT ALGORITHM FOR NEMATIC LIQUID CRYSTAL FLOW BASED ON THE GAUGE-UZAWA METHOD

In this paper,we present a finite element algorithm for the time-dependent nematic liquid crystal flow based on the Gauge-Uzawa method.This algorithm combines the Gauge and Uzawa methods within a finite element variational formulation,which is a fully discrete projection type algorithm,whereas many projection methods have been studied without space discretization.Besides,error estimates for velocity and molecular orientation of the nematic liquid crystal flow are shown.Finally,numerical results are given to show that the presented algorithm is reliable and confirm the theoretical analysis.

Residual-type a posteriori error estimate for parabolic obstacle problems

In this paper, a posteriori error estimates were derived for piecewise linear finite element approximations to parabolic obstacle problems. The instrumental ingredient was introduced as a new interpolation operator which has optimal approximation properties and preserves positivity. With the help of the interpolation operator the upper and lower bounds were obtained.

MIXED FINITE ELEMENT METHODS BASED ON RIESZ-REPRESENTING OPERATORS FOR THE SHELL PROBLEM

To solve the shell problem, we propose a mixed finite element method with bubble-stabili -zation term and discrete Riesz-representation operators. It is shown that this new method is coercive, implytng the well-known X-ellipticity and the Inf-Sup condition being circumvented, and the resulting linear system is symmetrically positively definite, with a condition number being at most O(h-2). Further, an optimal error bound is attained.

THE APPROXIMATIONS OF THE EXACT BOUNDARY CONDITION AT AN ARTIFICIAL BOUNDARY FOR LINEARIZED INCOMPRESSIBLE VISCOUS FLOWS

We consider the linearized incompressible Navier-Stokes (Oseen) equations in a flat channel. A sequence of approximations to the exact boundary condition at an artificial boundary is derived. Then the original problem is reduced to a boundary value problem in a bounded domain, which is well-posed. A finite element approximation on the bounded domain is given, furthermore the error estimate of the finite element approximation is obtained. Numerical example shows that our artificial boundary conditions are very effective.

Analysis and Numerical Approximation of an Electro-Elastic Frictional Contact Problem

We consider a mathematical model which describes the static frictional contact between a piezoelectric body and a conductive foundation.A non linear electro-elastic constitutive law is used to model the piezoelectric material.The unilateral contact is modelled using the Signorini condition,nonlocal Coulomb friction law with slip dependent friction coefficient and a regularized electrical conductivity condition.Existence and uniqueness of a weak solution is established.A finite elements approximation of the problem is presented,a priori error estimates of the solutions are derived and a convergent successive iteration technique is proposed.

A PRIORI ERROR ESTIMATE AND SUPERCONVERGENCE ANALYSIS FOR AN OPTIMAL CONTROL PROBLEM OF BILINEAR TYPE

In this paper, we investigate a priori error estimates and superconvergence properties for a model optimal control problem of bilinear type, which includes some parameter estimation application. The state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions. We derive a priori error estimates and superconvergence analysis for both the control and the state approximations. We also give the optimal L^2-norm error estimates and the almost optimal L^∞-norm estimates about the state and co-state. The results can be readily used for constructing a posteriori error estimators in adaptive finite element approximation of such optimal control problems.

RECOVERY A POSTERIORI ERROR ESTIMATES FOR GENERAL CONVEX ELLIPTIC OPTIMAL CONTROL PROBLEMS SUBJECT TO POINTWISE CONTROL CONSTRAINTS

Superconvergence and recovery a posteriori error estimates of the finite element ap- proximation for general convex optimal control problems are investigated in this paper. We obtain the superconvergence properties of finite element solutions, and by using the superconvergence results we get recovery a posteriori error estimates which are asymptotically exact under some regularity conditions. Some numerical examples are provided to verify the theoretical results.

A Parallel Computational Model for Three-Dimensional,Thermo-Mechanical Stokes Flow Simulations of Glaciers and Ice Sheets

This paper focuses on the development of an efficient,three-dimensional,thermo-mechanical,nonlinear-Stokes flow computational model for ice sheet simulation.The model is based on the parallel finite element model developed in[14]which features high-order accurate finite element discretizations on variable resolution grids.Here,we add an improved iterative solution method for treating the nonlinearity of the Stokes problem,a new high-order accurate finite element solver for the temperature equation,and a new conservative finite volume solver for handling mass conservation.The result is an accurate and efficient numerical model for thermo-mechanical glacier and ice-sheet simulations.We demonstrate the improved efficiency of the Stokes solver using the ISMIP-HOM Benchmark experiments and a realistic test case for the Greenland ice-sheet.We also apply our model to the EISMINT-II benchmark experiments and demonstrate stable thermo-mechanical ice sheet evolution on both structured and unstructured meshes.Notably,we find no evidence for the“cold spoke”instabilities observed for these same experiments when using finite difference,shallow-ice approximation models on structured grids.