Collocation Finite Element Method Research Articles

Finite element method for the solution of Maxwell's equations in multiple media

This paper concerns a subdomain collocation finite element method which is applied to solve the electromagnetic equations with two or more media. The solution is discontinuous. Both the implementation and the proof of convergence are discussed.

Finite element collocation method for variably saturated flow in two space dimensions : Allen, M B; Murphy, C L Water Resour ResV22, N11, Oct 1986, P1537–1542

Finite element collocation method for variably saturated flow in two space dimensions : Allen, M B; Murphy, C L Water Resour ResV22, N11, Oct 1986, P1537–1542

A finite element collocation method for variably saturated flows in porous media

AbstractOne common formulation of Richard's equation for variably saturated flows in porous media treats pressure head as the principal unknown and moisture content as a constitutive variable. Numerical approximations to this “head‐based” formulation often exhibit mass‐balance errors arising from inaccuracies in the temporal discretization. This article presents a finite‐element collocation scheme using a mass‐conserving formulation. The article also proposes a computable index of global mass balance.

A Finite Element Collocation Method for Bisingular Integral Equations

AMS (MOS): 45E05, 65R20, 45L05, 47D25 This paper presents a finite element collocation method for bisingular integral equations with continuous coefficients on the torus. The convergence properties of this method are proved by using a local principle and Banach algebra techniques.

A Collocation Finite Element Method for Integration of the Boundary Layer Equations

The author proposes a collocation finite element procedure using the Hermite basis functions. The particular choice of the trial functions gives a physical interpretation to all the nodal values. In order to demonstrate the versatility of collocation, three classical problems from boundary layer theory are solved numerically. The calculations verify that the method can produce accurate information at low cost.

AN APPLICATION OF THE FINITE ELEMENT COLLOCATION METHOD IN SOLVING SELECTED PROBLEMS OF SUBMICRON AEROSOL FILTRATION. THE CASE OF THE SLIP FLOW ON A FIBRE

Abstract The convective diffusion equation of aerosols in the case of gas stream slip on the fibre cylindrical surface has been solved by means of the finite element collocation method. Half-analytical form of the solution makes it possible to examine directly changes the slip causes in the distribution of solids around the fibre surface. Concentration profiles and deposition efficiency computed by this method differ from the experiment results less than those obtained by the common method of finite differences. Additionally the finite element collocation has proved to converge more rapidly than that of finite differences (at optimal conditions of calculation for both methods).10

A moving finite element collocation method for transient problems with steep gradients

A moving finite element collocation method for transient problems with steep gradients

C 1 — Collation and continuity requirements

This communication deals with an inherent difficulty related to continuity in C1 finite element collocation methods. It is shown that the C1 condition is too special. Consequently these methods, as usually developed, are too restrictive and not suited to general purpose finite element programs, even for two-point problems. In view of this elementary observation, it may be argued that this approach has received more attention than this form of the method really merits, and that other (weaker) collocation procedures are probably more worthy.

Solution of models for gassolid noncatalytic reactions by orthogonal collocation on finite elements with moving boundary

A variable size grain model is used to describe gassolid non-catalytic reactions which are accompanied by change in solid structure. A finite element collocation method with a moving boundary is developed to describe gas and solid concentration profiles in the reacting pellet and to compute conversion-time relationships. Such a method is necessary in case of structural changes which lead to complete pore mouth blockage and gives good results until such blockage occurs. A modified shrinking core model is used as an approximation after the time at which complete pore mouth closure occurs. The developed computational algorithms show that the effect of structural changes on time-conversion relationship is significant.

Collocation finite-element simulation of thermal-ice regimes in rivers

A collocation finite element scheme using Hermite cubic piecewise polynomials is developed for the numerical simulation of problems pertaining to one-dimensional convective diffusion processes. This scheme is tested and compared with a number of existing solutions, and is applied to simulate the winter thermal-ice regimes in natural rivers. The result of this study suggests that the collocation finite element method achieves accurate simulation with considerably less computer time than other well-known numerical methods.

Finite element collocation solution of tubular and packed-bed reactor two point boundary value problems

Nonlinear two point boundary value problems (TPBVP's) are frequently encountered in chemical engineering. TPBVP's describing tubular and packed-bed reactors are important and have received much attention in the literature. For certain ranges of parameter values of the reactors, the TPBVP's become stiff and pose problems in numerical solution. These problems have been efficiently solved by the finite element collocation method using Hermite and B-spline functions in conjunction with quasilinearization. Numerical results have been presented and discussed. The problems have been solved for a wide range of parameter values without encountering any numerical difficulty.

A $C^1 $ Finite Element Collocation Method for Elliptic Equations

Collocation at Gaussian quadrature points as a means of determining a $C^1 $ finite element approximation to the solution of a linear elliptic boundary value problem on a square is studied. Optimal order $L^2 $ and $H^1 $ error estimates are established for approximation in a function space consisting of tensorproducts of $C^1 $ piecewise polynomials of degree not greater that r, where $r \geqq 3$.

Collocation and galerkin finite element methods for viscoelastic fluid flow—II. Die swell problems with a free surface

The die swell problem occurs when a fluid flows down a capillary and is emitted into a gas which exerts negligible shear and normal forces on the jet. The die swell problem is solved for Newtonian and viscoelastic fluids. The viscoelastic fluids use a nonlinear Maxwell model but we report calculations for only a small elastic component. Both Galerkin and collocation Finite Element Methods are used and compared. The role of natural boundary conditions in the Galerkin method is compared to the pointwise satisfaction of boundary conditions in the collocation method.

Finite Element Collocation Methods for First Order Systems

Finite Element Collocation Methods for First Order Systems

Finite element collocation methods for first-order systems

Finite element methods and the associate collocation methods are considered for solving first-order hyperbolic systems, positive in the sense of Friedrichs. Applied in the case when the meshes are rectangle, those methods lead for example to the successfully used box scheme for the heat equation or D.S.N. scheme for the neutron transport equation. Generalizations of these methods are described here for nonrectangle meshes and (or) noncylindrical domains; stability results and error estimates are derived.

A collocation finite element method for potential problems in irregular domains

AbstractA potentially powerful numerical method for solving certain boundary value problems is developed. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees‐of‐freedom per node are used. A subparametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representation of irregular boundaries. It is shown that by taking advantage of the boundary conditions, a minimum number of collocation points can be used.The method is particularly suitable for potential and mass transport problems where a C1 continuous solution is required. In contrast to the Galerkin approach, it does not require the evaluation of basis function products and numerical integration, also the coefficient matrix contains only about half as many non‐zero terms as the corresponding Galerkin coefficient matrix. This results in approximately a 90 per cent reduction in formulation and a 50 per cent reduction in solution operation, as compared with the Galerkin finite element method, for this type of problem. Examples show that the accuracy of the collocation solution is as good as or better than that of the Galerkin solution.

A finite element collocation method for the exact control of a parabolic problem

AbstractA finite element method is given for the problem of exact control of a linear parabolic equation. The basis functions consist of piecewise bicubic polynomials and the differential equation is satisfied at Gaussian collocation points within each element. The overdetermined system of equations obtained is solved by the method of least squares, and a convergence argument is given for the complete procedure. Numerical results are given for two problems of boundary control.

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 Finite Element Collocation Method for Quasilinear Parabolic Equations

A Finite Element Collocation Method for Quasilinear Parabolic Equations

A finite element collocation method for quasilinear parabolic equations

Let the parabolic problem c ( x , t , u ) u t = a ( x , t , u ) u x x + b ( x , t , u , u x ) , 0 > x > 1 , 0 > t ≦ T , u ( x , 0 ) = f ( x ) , u ( 0 , t ) = g 0 ( t ) , u ( 1 , t ) = g 1 ( t ) c(x,t,u){u_t} = a(x,t,u){u_{xx}} + b(x,t,u,{u_x}),0 > x > 1,0 > t \leqq T,u(x,0) = f(x),u(0,t) = {g_0}(t),u(1,t) = {g_1}(t) , be solved approximately by the continuous-time collocation process based on having the differential equation satisfied at Gaussian points ξ i , 1 {\xi _{i,1}} and ξ i , 2 {\xi _{i,2}} in subintervals ( x i − 1 , x i ) ({x_{i - 1}},{x_i}) for a function U : [ 0 , T ] → H 3 U:[0,T] \to {\mathcal {H}_3} , the class of Hermite piecewise-cubic polynomial functions with knots 0 = x 0 > x 1 > ⋯ > x n = 1 0 = {x_0} > {x_1} > \cdots > {x_n} = 1 . It is shown that u − U = O ( h 4 ) u - U = O({h^4}) uniformly in x and t, where h = max ( x i − x i − 1 ) h = \max ({x_i} - {x_{i - 1}}) .