ERROR ANALYSIS OF THE P-VERSION DISCONTINUOUS GALERKIN METHOD FOR HEAT TRANSFER IN BUILT-UP STRUCTURES

In this paper, we provide an error analysis for the $p$-version of 
the discontinuous Galerkin finite element method for a class of 
heat transfer problems in built-up structures. Also, a general 
form of the matrix associated with the discretization of time 
variable using the $p$-finite element basis functions is 
established. Many interesting properties of this matrix are 
obtained. Numerical examples are provided in the last section.


Introduction.
The purposes of this paper are to report the state of the art information on time discretization techniques in the discontinuous Galerkin method for parabolic problems (this section) and to establish error analysis for pversion of the finite element method for such problems (section 2). Also a discussion of various time discretization techniques are included (section 3). The discontinuous Galerkin method is applied to the following standard model problem of parabolic type: Find u such that where R is a closed and bounded set in R3 with boundary 80, Rf = (0, GO), Au = d2u/dx2 + d2/dy2 + d2u/dz2, ut = &/at, and the functions f and ~0 are given data. In this paper, region 0 is assumed to be a thin body in R3, such as a panel on the wing or fuselage of an aerospace vehicle. pversion of the finite element method is considered in all directions including time variable. Because of the special characteristics of the region, it. is assumed that, through the of all polynomials of degree 5 p with coefficients in X , i.e., q ( t ) E P p ( I ; X ) if and only if q ( t ) = C;==oxjtj for some xj E X and t E I . Let TI be a partition of I into M ( I ) subintervals and it is, of course, equal to C,M_(:)(p, + 1). The semidiscrete solution U E PPn(In;X) of the problem ( l . l ) , if U is already determined on Ik, 1 5 k 5 n -1, is found by solving the following problem: Find U E PPn(In; X ) such that for all V E PPn(In; X ) and UF = 210.
Here we assume that L2(Cl) is densely embedded in a Banach space X . The following theorem and its subsequent corollary are reported in [lo]. Theorem describes the error estimate for the semidiscrete solution explicitly in terms of the time steps, the approximation orders and the local regularities of the solution.
Theorem 1.1 Let u be the solution of (1.1) and U the semidiscrete solution in Vp(T1;X).

integers. Then
In the pversion of the discontinuous Galerkin finite element method, a temporal partition TI is fixed and convergence is obtained by p , --f 30. For p, = p for each n = 1. . . . , h.i (I). and for a smooth solution u, we obtain the following: Remark 1.1: As pointed out in [lo], Corollary 1.2 shows that for smooth solutions where SO is large, it is better to increase p than t,o reduce I C, , , at a k e d , often low p. Since N NRDOF(VP(T1; X ) )p , we see that for pversion of the finite element method, Using the standard approximation theory for analytic functions (1.3) reduces to for some b > 0 independent of p . If the solution is not smooth in time, it is still possible to approximate it in exponential orders by a hpfinite element method which combines a certain geometric partition with the semidiscrete space VP(TI; X ) where p is linearly distributed, (see [lo] for detail). Using the h-finite element method with non-uniform graded time partitions, such non-smooth solutions can be approximated in an algebraically optimal order, but not exponentially, using different approaches, (see, e.g., 151, [lo]). The standard pfinite element method does not perform well in this context. Therefore, since we aim to establish the pversion of the finite element method for ( l . l ) , we assume for the remaining of this paper that solutions are smooth in time. This assumption allows us to establish the order of approximation in time variable that is compatible with the approximation orders in the spatial variables.
Now we consider the problem of discretizing the space 0. For simplicity, we assume that  In the z-variable for through the thickness approximation, the local variable r is defined in the reference element [-1,1] and r is mapped onto the reference element by Qz, i.e., Clearly, QL is a linear function defined by The Jacobian of Qz is constant In this paper, the basis functions of Pp([-l, 11) are taken to be the one-dimensional hierarchical shape functions. See [8] for a complete discussion of the basis elements used in the p and hpfinite element methods. Note that $i'~ form an orthogonal family with respect to the energy inner product (., . ) E , -1 Also note that the internal shape functions satisfy For the case 1 = 2 and p 2 3, the four nodal shape functions and the remaining p -3 internal shape functions given by In this case, the internal shape functions satisfy with h = maxK,T,, diam(K). and for some y > 0. PP(r) denotes the space of all polynomials of degree 5 p defined on r. and 1.4, the following is proved in [5]: where vi = min(k,pi + 1) for i = 1,2 and h = maxKET,, diam(K), with Th a triangulation of w.
A straightforward extension of Theorem 1.5 provides the following which describes a total error where C1, C 2 and C3 are constants independent of h, k, , , , d, p l , p2 and p3.
Using (1.4), we obtain where C and b are independent of p17p2 and p3.

p-Version of Discontinuous Galerkin Finite Element Method
In this section, pversion of discontinuous Galerkin finite element method for the parabolic problem (1.1) is described. The main goal here is to provide an error analysis for the pfinite element method using the results from Section 1. The semidiscrete approximation equation (1.2) is now upgraded to a fully discretized equat,ion below. It is assumed as in Theorem 1.5 and Corollary 1.6 that the degree vectors in space, through-the-thickness and time are assumed to be p l , pa and p3 respectively. Define

W(Pl?PZJ'3) = { V : R +
Then the fully discretized discontinuous Galerkin method can be described as follows: Find U E W@'>mJQ) such that for n = 1,2,. . . , We consider in (1.1) only the case of isotropic materials along with Dirichlet boundary conditions. However, extensions to anisotropic materials as well as mixed boundary conditions where Neumann boundary conditions are incorporated are possible and the present analysis carries over to these cases. The thesis by Tomey [7] treats transversely anisotropic materials as well as isotropic materials along with a mixed boundary condition.

Discretizations in Time Variable.
In this section, effects of the use of different basis elements t o approximate the solution in time variable are considered. The solution u is approximated over K1 x (-g, $) x I, using the outer tensor products: U I~~~( -~,~,~~~ = (4 8 1c, B q T a n = 3 a n .  The set of the canonical polynomials 8,+1(t) = tV were used in [7] and the components of the matrices which represent A i j and B i j were computed exactly. We consider two other alternatives for 8. ... and 1

(3.14)
It is reported in Ill] that, as the result of their numerical experiments, the matrix Ap+l is diagonalizable up to its order p = 100. Subsequently, equation (3.13) is decoupled via (3.14) into p + 1 independent scalar equations, each of which requires complex arithmetic to solve. A new approach which uses the real Schur decomposition is now presented. The new approach does not require the complex arithmetic. where each &i is eith,er a 1 x 1 matrix or a 2 x 2 matrix having complex conjugate eigenvalues of A .
By Theorem 3.1, it is guaranteed that no 1 x 1 matrix in Schur decomposition for A is 0.
Also, it is worth noting that the Schur decomposition is an orthogonal similarity transformation and thus avoid the computation of Q-l as required in the diagonalization process done in

-2
Fp-2 2 The case for 1 x 1 krn is similar. It is important to recall that the computation thus described can be completed because of Theorem 3.1.

Start-up Singularities:
In this final section, we make some comments on the start-up singularities normally associated with the parabolic problems. The regularity assumptions in Theorem 2.1 and Corollary 2.2 were taken so that the current fully pfinite element method could provide numerical solutions where the discretization error associated in time can be made consistent with the discretization error associated in space. However. as indicated earlier. time singularities arise due to various types of incompatible initial data. To capture such singularities, hpversion of finite element method must be considered. In [lo], a nonuniform time discretization is determined by considering the conditions on f as well as the initial function uo in (1.1). More specifically, the function f is assumed to be piecewise analytic as a function on [0, T ] with values in H , Le., with constants C and d independent of 1 and t. Also, uo is assumed to be in He = (H,X)e,2, 0 5 0 5 1, where ( H , X)e,2 is a space between H and X determined by the K-method of interpolation, (see [8]). An h-version of the discontinuous Galerkin finite element method developed in [5] establishes a nonuniform time discretization scheme which is based upon the behavior of with parabolic problems arise. Analysis used in determining the nonuniform graded partition points in [5] is distinct from the one used in [lo] and an example is provided in [ 5 ] , which demonstrates that the method of Kaneko, Bey and Hou gives more sparse time partition points than the ones given in [lo].