A note on the design of hp-adaptive finite element methods for elliptic partial differential equations

We introduce an hp-adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp-extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local Legendre series expansion. Numerical experiments confirm the robustness and reliability of the proposed algorithm.     

An electric-analog simulation of elliptic partial differential equations using finite element theory

Elliptic partial differential equations can be solved using the Galerkin-finite element method to generate the approximating algebraic equations, and an electrical network to solve the resulting matrices. Some element configurations require the use of networks containing negative resistances which, while physically realizable, are more expensive and time-consuming to construct.     

Reduced-basis output bounds for approximately parametrized elliptic coercive partial differential equations

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with (approximately) affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space W_N spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N, typically very small, and the (approximate) parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.In our earlier work, we develop a rigorous a posteriori error bound framework for the case in which the parametrization of the partial differential equation is exact; in this paper, we address the situation in which our mathematical model is not complete. In particular, we permit error in the data that define our partial differential operator: this error may be introduced, for example, by imperfect specification, measurement, calculation, or parametric expansion of a coefficient function. We develop both accurate predictions for the outputs of interest and associated rigorous a posteriori error bounds; and the latter incorporate both numerical discretization and model truncation effects. Numerical results are presented for a particular instantiation in which the model error originates in the (approximately) prescribed velocity field associated with a three-dimensional convection-diffusion problem.     

A new parallel domain decomposition method for the adaptive finite element solution of elliptic partial differential equations

We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain—although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in 15 , it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus allowing the accuracy, reliability and efficiency of mesh adaptivity to be utilized in a well load-balanced manner. Finally, numerical evidence is presented which suggests that this technique has significant potential, both in terms of the rapid convergence properties and the efficiency of the parallel implementation. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite element approximation of elliptic partial differential equations on implicit surfaces

The aim of this paper is to investigate finite element methods for the solution of elliptic partial differential equations on implicitly defined surfaces. The problem of solving such equations without triangulating surfaces is of increasing importance in various applications, and their discretization has recently been investigated in the framework of finite difference methods. For the two most frequently used implicit representations of surfaces, namely level set methods and phase-field methods, we discuss the construction of finite element schemes, the solution of the arising discretized problems, and provide error estimates. The convergence properties of the finite element methods are illustrated by computations for several test problems.     

L 2-Error bounds for approximate solutions of elliptic partial differential equations with Dirichlet-boundary conditions

New a posteriori (computable) upper bounds for theL ₂-norms, both ofD(u?v) and ofu?v are proposed, whereu is the exact solution of the boundary value problem $$Au: = - D(pDu) + qu = f, x \in G and u = 0,x \in \partial G$$ andv any approximation of it (D is here the vector of partial derivatives with respect to the components ofx). It is shown that the new error bounds are better than the classical one, which is proportional to ‖Av?f‖, in many cases. This happens, e. g., ifq has some zero point inG, as in the case of a Poisson equation.     

A complex variable boundary element method for elliptic partial differential equations in a multiple-connected region

A simple boundary element method based on the Cauchy integral formulae is proposed for the numerical solution of a class of boundary value problems involving a system of elliptic partial differential equations in a multiple-connected region of infinite extent. It can be easily and efficiently implemented on the computer.     

A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in $H^1$-norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.     

Multilevel Schwarz methods for elliptic partial differential equations

We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method. In our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary value problems, including a convection–diffusion problem when suitable stabilization becomes necessary.     

On solving elliptic stochastic partial differential equations

A model elliptic boundary value problem of second order, with stochastic coefficients described by the Karhunen–Loève expansion is addressed. This problem is transformed into an equivalent deterministic one. The perturbation method and the method of successive approximations is analyzed. Rigorous error estimates in the framework of Sobolev spaces are given.     

Explicit group over-relaxation methods for solving elliptic partial differential equations

Previous block (or line) iterative methods have been implicit in nature where a group of equations (or points on the grid mesh) are treated implicitly [2] and solved directly by a specialised algorithm, this has become the standard technique for solving the sparse linear systems derived from the discretisation of self-adjoint elliptic partial differential equations by finite difference/element techniques.The aim of this paper is to show that if a small group of points (i.e. 2, 4, 9, 16 or 25 point group) is chosen then each group can easily be initially inverted leading to a new class of Group Explicit iterative methods. A comparison with the usual 1-line and 2-line block S.O.R. schemes for the model problem confirm the new techniques to be computationally superior.     

A posteriori error analysis of space-time finite element discretizations of the time-dependent Stokes equations

We present a novel a posteriori error analysis of space-time finite element discretizations of the time-dependent Stokes equations. Our analysis is based on the equivalence of error and residual and a suitable decomposition of the residual into spatial and temporal contributions. In contrast to existing results we directly bound the error of the full space-time discretization and do not resort to auxiliary semi-discretizations. We thus obtain sharper bounds. Moreover the present analysis covers a wider range of discretizations both with respect to time and to space.     

On finite element methods for elliptic equations on domains with corners

A finite element method for approximating elliptic equations on domains with corners is proposed. The method makes use of the singular functions of the problem in the trial space and the kernel functions of the adjoint problem in the test space. This leads to good approximates of the coefficients of the singular functions. In the numerical computations, the method is compared with the well known Singular Function Method.     

Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization.     

Primal hybrid finite element methods for 4th order elliptic equations

A primal hybrid method for the biharmonic problem is developed. We find convergence results for a large class of approximations. The associated non conforming elements prove to pass ahigher order patch test and have the optimal order of convergence.     

An incomplete nested dissection algorithm for parallel direct solution of finite element discretizations of partial differential equations

A multilevel algorithm is presented for direct, parallel factorization of the large sparse matrices that arise from finite element and spectral element discretization of elliptic partial differential equations. Incomplete nested dissection and domain decomposition are used to distribute the domain among the processors and to organize the matrix into sections in which pivoting is applied to stabilize the factorization of indefinite equation sets. The algorithm is highly parallel and memory efficient; the efficient use of sparsity in the matrix allows the solution of larger problems as the number of processors is increased, and minimizes computations as well as the number and volume of communications among the processors. The number of messages and the total volume of messages passed during factorization, which are used as measures of algorithm efficiency, are reduced significantly compared to other algorithms. Factorization times are low and speedups high for implementation on an Intel iPSC/860 hypercube computer. Furthermore, the timings for forward and back substitutions are more than an order-of-magnitude smaller than the matrix decomposition times.     

Development,evaluation and selection of methods for elliptic partial differential equations

We present a framework within which to evaluate and compare computational methods to solve elliptic partial differential equations. We then report on the results of comparisons of some classical methods as well as a new one presented here. Our main motivation is the belief that the standard finite difference methods are almost always inferior for solving elliptic problems and our results are strong evidence that this is true. The superior methods are higher order (fourth or more instead of second) and we describe a new collocation finite element method which we believe is more efficient and flexible than the other well known methods, e.g., fourth order finite differences, fourth order finite element methods of Galerkin, Rayleigh-Ritz or least squares type.Our comparisons are in the context of the relatively complicated problems that arise in realistic applications. Our conclusion does not hold for simple model problems (e.g., Laplaces equation on a rectangle) where very specialized methods are superior to the generally applicable methods that we consider. The accurate and relatively simple treatment of boundary conditions involving curves and derivations is a feature of our collocation method.     

A posteriori error estimates¶for nonconforming finite element schemes

We derive a posteriori error estimates for nonconforming discretizations of Poisson's and Stokes' equations. The estimates are residual based and make use of weight factors obtained by a duality argument. Crouzeix-Raviart elements on triangles and rotated bilinear elements are considered. The quadrilateral case involves the introduction of additional local trial functions. We show that their influence is of higher order and that they can be neglected. The validity of the estimate is demonstrated by computations for the Laplacian and for Stokes' equations. Received: November 1998 / Accepted: January 1999