Optimal L∞-error estimates for nonconforming and mixed finite element methods of lowest order

Summary For second order linear elliptic problems, it is proved that theP ₁-nonconforming finite element method has the sameL -asymptotic accuracy as theP ₁-conforming one. This result is applied to derive optimalL -error estimates for both the displacement and the stress fields of the lowest order Raviart-Thomas mixed finite element method, and a superconvergence result at the barycenter of each element.Performed in the research program of Istituto di Analisi Numerica of C.N.R. of PaviaPartially supported by MPI, GNIM of CNR, ItalySupported by Consejo Nacional de Investigaciones Cientificas y Técnicas, Argentina     

L ∞-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations

We look at L ^∞-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations. In so doing, use is made of mixed finite element methods. The state and costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control, by piecewise constant functions. L ^∞-error estimates of optimal order are derived for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, numerical tests are presented which confirm our theoretical results.     

SharpL ∞-error estimates for semilinear elliptic problems with free boundaries

Summary A numerical scheme to approximate a semilinear PDE involving a (singular) maximal monotone graph is analyzed inL . A preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute; the discrete maximum principle is thus avoided. Sharp pointwise error estimates are derived for both the smoothing and the discretization procedures. An optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence. These error estimates for solutions, in conjunction with nondegeneracy properties of continuous problems, provide sharp interface error estimates. Two model examples are discussed: the obstacle problem and a combustion equation.This work was partially supported by Consiglio Nazionale delle Ricerche of Italy while the author was in residence at the Istituto di Analisi Numerica del C.N.R. di Pavia     

Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems

In this paper we prove the uniform convergence of the standard multigrid V-cycle algorithm with the Gauss-Seidel relaxation performed only on the new nodes and their "immediate" neighbors for discrete elliptic problems on the adaptively refined finite element meshes using the newest vertex bisection algorithm. The proof depends on sharp estimates on the relationship of local mesh sizes and a new stability estimate for the space decomposition based on the Scott-Zhang interpolation operator. Extensive numerical results are reported, which confirm the theoretical analysis.     

L ∞ andW 1, ∞ estimates for second order generalized difference methods on two-point boundary value problems

In this paper, we consider the second order generalized difference scheme for the two-point boundary value problem and obtain optimal order error estimates inL ^∞ andW ^{1, ∞}. The results in this paper perfect the theory of the second order generalized difference method.     

High order finite volume methods for singular perturbation problems

In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.     

A posteriori error estimates of finite element method for parabolic problems

1IntroductionThebase0fadaPtivecomputing0ffiniteelementmethodisap0steri0rierr0restimates.I.Babuskaisthepioneerinthisfields.Manytechniquesaredevel0pedtoobtainaposteri0rierrorestimators.See[1-3,7-8,19-201.Theyaremainlybased0nthejumps0fthederiva-tivesontheboundariesoftl1eelel11elltandtheresidualintheelemellts.Recelltresultssh0wthatthereareveryclosedrelatiollsbetweellasymptoticexactap0steri0rierrorestimatesandsuperc0nvergence-SeealsoQ.Linetal.[11-13],andChen-Huang['].Therehasbeenmuchprogressill…     

New a posteriori L∞(L2) and L2(L2)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L^∞(J; L²Ω)-norm and L²(J; L²Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.     

Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms

Two residual-based a posteriori error estimators of the nonconforming Crouzeix-Raviart element are derived for elliptic problems with Dirac delta source terms.One estimator is shown to be reliable and efficient,which yields global upper and lower bounds for the error in piecewise W1,p seminorm.The other one is proved to give a global upper bound of the error in Lp-norm.By taking the two estimators as refinement indicators,adaptive algorithms are suggested,which are experimentally shown to attain optimal convergence orders.     

A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

A remark on least-squares mixed element methods for reaction–diffusion problems

In this paper, we propose a least-squares mixed element procedure for a reaction–diffusion problem based on the first-order system. By selecting the least-squares functional properly, the resulting procedure can be split into two independent symmetric positive definite schemes, one of which is for the unknown variable and the other of which is for the unknown flux variable, which lead to the optimal order H¹(Ω)$H^1(\Omega )$ and L²(Ω)$L^2(\Omega )$ norm error estimates for the primal unknown and optimal H(div;Ω)$H(\mathrm{div};\Omega )$ norm error estimate for the unknown flux. Finally, we give some numerical examples.     

Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems

In this paper, a second order modified method of characteristics defect-correction (SOMMOCDC) mixed finite element method for the time dependent Navier–Stokes problems is presented. In this method, the hyperbolic part (the temporal and advection term) are treated by a second order characteristics tracking scheme, and the non-linear term is linearized at the same time. Then, we solve the equations with an added artificial viscosity term and correct this solution by using the defect-correction technique. The error analysis shows that this method has a good convergence property. In order to show the efficiency of the SOMMOCDC mixed finite element method, we first present some numerical results of an analytical solution problem, which agrees very well with our theoretical results. Then, we give some numerical results of lid-driven cavity flow with the Reynolds number Re = 5,000, 7,500 and 10,000. From these numerical results, we can see that the schemes can result in good accuracy, which shows that this method is highly efficient.     

L∞-boundedness of the finite element galerkin operator for parabolic problems

In this paper the heat equation with Dirichlet boundary conditions in N ≤ 3 space dimensions - serving as model problem of second order parabolic initial boundary value problems - is considered. We prove: The standard finite element method is uniformly bounded in L_∞ with respect to space and time if the underlying finite elements are at least cubics.     

OptimalL ∞-error estimates for piecewise linear finite elements inR n for 1≦n≦3

Summary In this paper, for two second order elliptic boundary value problems, uniform convergence error estimates are derived using piecewise linear (conforming and non-conforming) finite elements. The estimates are valid for polyhedral convex regions R ⁿ with 1n3. The proof of the estimates is mainly based on Sobolev's inequality, by which the uniform estimates are reduced to estimates in the mean. In this way, it is possible to obtain the desired uniformO(h ²)-estimate in the plane as a special case, thus confirming theoretically Natterer's numerical results.

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Zusammenfassung In dieser Arbeit werden für zwei elliptische Randwertprobleme zweiter Ordnung Fehlerabschätzungen in der Maximumnorm hergeleitet, wenn die finiten Elemente stückweise linear (konform oder nichtkonform) sind. Die Fehlerabschätzungen gelten für konvexe polyederförmige Gebiete R ⁿ mit 1n3. Der Beweis für die Fehlerabschätzungen beruht hauptsächlich auf der Sobolewschen Ungleichung, mit deren Hilfe die gleichmäßigen Abschätzungen auf Abschätzungen im Mittel zurückgeführt werden. Auf diese Weise erhält man für den Spezialfall der Ebene die gewünschte gleichmäßigeO(h ²)-Abschätzung, womit Natterer's numerische Ergebnisse bestätigt werden.

     

A second order nonconforming rectangular finite element method for approximating Maxwell’s equations

The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell’s equations.Then the corresponding optimal error estimates are derived.The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O(h 3 ) ,properly one order higher than that of its interpolation error O(h 2 ) in the broken energy norm,where h is the subdivision parameter tending to zero.     

L ∞ stability of finite element approximations to elliptic gradient equations

Summary We examine theL stability of piecewise linear finite element approximationsU to the solutionu to elliptic gradient equations of the form –·[a(x)u]+f(x, u)=g(x) wheref is monotonically increasing inu. We identify a prioriL bounds for the finite element solutionU, which we call reduced bounds, and which are marginally weaker than those for the original differential equations. For the general,N-dimensionai, case we identify new conditions on the mesh, such that under the assumption thatf is Lipschitz continuous on a finite interval,U satisfies the reducedL bounds mentioned above. The new,N-dimensional regularity conditions preclude quasi-rectangular meshes.Moreover, we show thatU is stable inL in two dimensions for a discretization mesh on which –·[a(x)u] gives rise to anM-matrix, whileU is stable for any mesh in one dimension. The condition that the discretization of –·[a(x)u] has to be anM-matrix, still allows the inclusion of the important case of triangulating in a quasi-rectangular fashion.The results are valid for either the pure Neumann problem or the general mixed Dirichlet-Neumann boundary value problem, while interfaces may be present. The boundary conditions forU are obtained by use of (nonexpansive) pointwise projection operators.The first author is supported by the National Science Foundation under grant EET-8719100Research of the second author supported by National Science Foundation grant DMS.8420192