Advances in Mathematical Physics

Volume 2016, Article ID 7058017, 12 pages

http://dx.doi.org/10.1155/2016/7058017

## The Ritz Method for Boundary Problems with Essential Conditions as Constraints

^{1}Systems, Implementation & Integration, Smith Bits, A Schlumberger Co., 1310 Rankin Road, Houston, TX 77032, USA^{2}Computer and Mathematical Sciences, University of Houston-Downtown, One Main Street #S705, Houston, TX 77002, USA

Received 22 November 2015; Accepted 17 February 2016

Academic Editor: Pavel Kurasov

Copyright © 2016 Vojin Jovanovic and Sergiy Koshkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions. As in the Babuška-Brezzi approach boundary conditions are treated as variational constraints and Lagrange multipliers are used to remove them. However, we avoid the saddle point reformulation of the problem and therefore do not have to deal with the Babuška-Brezzi inf-sup condition. In higher dimensions boundary weights are used to approximate the boundary conditions, and the assumptions in our convergence proof are stated in terms of completeness of the trial functions and of the boundary weights. These assumptions are much more straightforward to verify than the Babuška-Brezzi condition. We also discuss limitations of the method and implementation issues that follow from our analysis and examine a number of examples, both analytic and numerical.

#### 1. Introduction

In variational problems linear boundary conditions are often divided into essential (geometric) and natural (dynamic) [1, II.12] and [2, 4.4.7]. More generally, one calls the boundary conditions essential if they involve derivatives of order less than half of the order of the differential equation and natural otherwise [3, I.1.2]. In the standard exposition of the Ritz method the trial functions may violate the natural conditions but must satisfy all the essential ones [2, 4.4.7] and [4]. The reason is that the variational equations force the natural conditions on the trial solutions anyway, even if the trial functions themselves do not satisfy them.

But what if we wish to use trial functions that violate the essential conditions as well? For instance, in problems involving parametric asymptotics, the trial functions are preimposed with no regard for boundary conditions [5, 6] and in initial-boundary problems with time-dependent boundary conditions the (time independent) trial functions can not satisfy them in principle. One may also wish to use such violating trial functions because they are simpler; see [7] for other possible reasons. Thus, there is abundant motivation to generalize the Ritz method to trial functions that do not satisfy the essential conditions. From a theoretical viewpoint this is a particular case of approximating solutions by nonconforming functions, the nonconformity here being at the boundary [8].

A natural idea is to treat the essential boundary conditions as variational constraints and to remove them as any other constraints using the Lagrange multipliers. Such approach is naively taken in some applied works at least since 1946 [9]; see also [10] where the authors explicitly cite the simplicity of the trial functions as a reason for using them. Babuška et al. [8, Sec.7] and [11] were the first to theoretically analyse the use of Lagrange multipliers in the context of nonconforming Finite Element Method (FEM); their work was generalized to more general trial functions by Brezzi [12] and [13, II.1]. Their analysis relies on the saddle point reformulation of the original variational problem and leads to the celebrated Babuška-Brezzi inf-sup condition that dictates a strict relation between choices of spaces for the trial functions and for the Lagrange multipliers; see also [13, 14] for applications in non-FEM context. This approach however is very far from the intuitive reasoning behind the naive application of the Lagrange multipliers in [9] or in [10]. As a result, the meaning of the Babuška-Brezzi condition remains obscure, which explains why it remains relatively unfamiliar to nonexperts, and its verification is often quite involved mathematically [14]. The standard approach applies to problems with quadratic functionals [13, II.1] but produces very strong stability and approximation results.

It is not our purpose to match the technical sophistication of the Babuška-Brezzi approach, but to give a more elementary and natural derivation of the Ritz method with Lagrange multipliers, the Ritz-Lagrange method for short. It applies to general convex functionals and is much closer to the standard theory of the Ritz method [15–17]. We avoid the saddle point reformulation altogether and work directly with the original variational problem. A system of boundary weights is used to approximate the Lagrange multipliers, and our assumptions are stated in terms of completeness of the trial functions and of the boundary weights. While completeness is implicitly contained in the Babuška-Brezzi condition, its role is by no means obvious or well-known, see [18], and it has significant practical consequences. On the other hand, completeness is much more intuitive and straightforward to verify than an inf-sup condition. As a trade-off, we only prove convergence of the method rather than obtain an explicit error estimate, so the relation between spaces of the trial functions and of the boundary weights is less precise. But hopefully a more intuitive approach provides a better understanding of analytic and numerical issues involved.

This paper is largely inspired by observations in [18] on intriguing and counterintuitive effects that the lack of completeness has on convergence of trial solutions to a true solution. We illustrate these effects further in a number of examples and explain them in the general context of convex analysis. Our approach leads to a number of practically useful observations. In particular, extra care is needed compared to the usual Ritz method: the variational functional has to be more regular on a larger space, the trial functions have to be complete in this larger space as well, and the multipliers can not be eliminated from the approximating systems using the usual variational formulas because of convergence issues. In the higher dimensional problems one needs to balance the numbers of the boundary weights and the trial functions to obtain well-posed approximating problems. While we do not prescribe this balance precisely, which would involve an analog of the Babuška-Brezzi condition, we still obtain a practical rule of thumb that works well in examples.

In Section 2 we introduce the Ritz-Lagrange method using simple one-dimensional examples, where not only the exact solution, but even all trial solutions are computed analytically. We also introduce notation and terminology of convex analysis needed to analyse the method theoretically. The proof of convergence is obtained in this case by straightforward reduction to the classical Ritz method. Unfortunately, this direct approach does not carry over to higher dimensions, and we develop a suitable generalization in Section 3. Numerical applications to multidimensional problems follow in Section 4. The paper ends with Conclusions, where we summarize our findings and discuss Galerkin type generalizations. Technical proofs are collected in the Appendix.

#### 2. Boundary Conditions as Variational Constraints

As a motivation, consider a boundary value problem for the second-order equation on with essential boundary conditions on both ends of the interval . We set for convenience , and to make everything explicitly computable. The exact solution is easily found to be . The corresponding variational functional isand the boundary value problem is equivalent to minimizing it on functions satisfying the boundary conditions.

We select , as our trial functions; they obviously do not satisfy the boundary conditions. Taking of them the trial solution is of the form with unknown coefficients ( in front of is for agreement with the convention for the cosine series). Since our boundary conditions are essential, and our trial functions do not satisfy them, the Ritz method has to be modified in some way. Our approach is to treat the essential conditions as variational constraints and remove them using Lagrange multipliers. The Lagrange functional is , where , are the Lagrange multipliers. Substituting, we find thatThe variational equations are , and adding the two boundary conditions we get the Ritz-Lagrange system. Solving these equations for unknowns , , and we get , for and all , while where is the floor function returning the largest integer not exceeding its argument. Thus, the trial solutions converge to the sum of the series: By extending the exact solution to an even function on and expanding it into a cosine series with coefficients [19, 12.1], one finds that is exactly the cosine series of the exact solution .

Variational problems typically come with a natural energy space, where convergence of solutions is considered [18]. On its energy space a variational functional typically has two important properties: it is continuous and it is weakly coercive; that is, [15, 6.2] and [16, III.10.2]. For a convex functional (we will only consider those) these two properties are sufficient to prove convergence of the usual Ritz approximations in the energy norm [15, 6.2A]. For the functional from (1) the energy space is , which is the Hilbert space of functions square integrable with their first derivatives and vanishing at and , with the norm . This norm is stronger than the norm in the sense that any convergent sequence converges in , but not conversely.

For our purposes the concept of energy space is not quite suitable because it incorporates the essential boundary conditions, and our trial functions do not satisfy them. Instead we start with a reflexive Banach space (the reader may assume it to be Hilbert without much loss) that has nothing to do with the boundary conditions, and a convex functional on it. Next, we introduce the boundary operator, a linear map , that maps functions to their boundary values. The subspace consists of functions that satisfy the boundary conditions. In our example we have , , and . The following three assumptions turn into a generalized analog of the energy space:(1) is convex and continuous.(2) on ; that is, is weakly coercive on .(3) is linear and continuous. This setup applies to homogeneous boundary conditions only. Nonhomogeneous conditions can be accommodated by selecting a function that satisfies them and switching to considering the differences with it. These differences solve the corresponding homogeneous problem, and all convergence issues can be reduced to them; see, for example, [20, 2.1].

We now turn to the trial functions. Recall that a system of elements in a Banach space is called complete if any element can be approximated by their linear combinations in the norm of the space. Let be a complete system in and let denote the linear span of . The Ritz-Lagrange approach amounts to minimizing on subject to the boundary conditions, which is equivalent to minimizing it on . Although by assumption about completeness all elements in can be approximated by linear combinations of , it is not a priori clear that functions from can be approximated by linear combinations of that are themselves in . The next lemma proved in the Appendix assures us that this is the case.

Lemma 1. *For any complete system of elements in there exists a system of their finite linear combinations belonging to , which is complete in .*

This lemma effectively reduces the Ritz-Lagrange method to the traditional Ritz method. Indeed, if is the complete system in produced by Lemma 1, then applying the Ritz method with as the trial functions amounts to minimizing on . In other words, the Ritz-Lagrange method with produces the same (up to reindexing) as the Ritz method with . This allows us to use well-known results on convergence of the Ritz method [15, 6.2A], [16, IV.12.4], and [21, 42.5] to prove convergence of its Ritz-Lagrange generalization. Producing the Ritz system involves differentiating the functional, so in addition to convexity and continuity we also have to assume its Gateaux differentiability [16, I.2.1], and [15, 3.2].

Theorem 2. *Let be a convex continuous Gateaux differentiable functional, let be a bounded linear operator, and let be a complete system in . If is weakly coercive on , then it has a minimizer on it, as well as minimizers on all , and there exists a subsequence such that , where denotes weak convergence in . Moreover, if is strictly convex on , then are unique and . In both cases the values of converge to its minimum on .*

The proof is fairly standard, but we outline it in the Appendix for the convenience of the reader. For general convex functionals only weak convergence can be expected; we discuss a stronger convexity assumption that guarantees convergence by norm in the next section. Note also that weak convergence in implies convergence by norm in due to Sobolev embedding theorems [20, I.6] and [22, 1.8].

Theorem 2 mostly justifies the Ritz-Lagrange method used to solve our example. The required properties of and the boundary operator are easily verified, except for the strict convexity, which follows from the Poincaré-Friedrichs inequality [20, I.6]. The completeness is a trickier issue. Completeness in follows from the standard theorems on cosine series [19, Ch.12], but we need a stronger form of completeness in . Fortunately, completeness of cosines reduces to the completeness of sines. The minimality below means that the system becomes incomplete after deleting any function.

Lemma 3. *The system , , is complete and minimal in .*

The proof is given in the Appendix.

As emphasized in [18] completeness is not a mere technicality in this context; it imposes a practical restriction on the choice of trial functions. To underscore the point, consider the biharmonic equation with the boundary conditions . For the exact solution is . If the cosine system is used again, proceeding as above we find thatThis time the trial solutions do not converge to the exact one; in fact . This is because* cosines are incomplete in *. As observed in [18], the second derivatives do not include a constant and therefore can not approximate the second derivative of in . But then cosines can not approximate in since its norm incorporates the norm for second derivatives. In [18] the authors add to the cosine system, but as we just saw the limit difference is a linear combination of and , so at least also needs to be added. We show in the Appendix that nothing else is needed.

Lemma 4. *The system , , , , is complete and minimal in .*

Note that* verifying completeness in a correct space can not be avoided even if one uses the usual Ritz method with trial functions satisfying the essential boundary conditions*. The second example demonstrates that solutions can not always be approximated in the space where the trial functions happen to be complete. Only completeness in the norm dictated by the variational functional counts.* Completeness of trial functions in a norm weaker than the energy norm does not simply weaken the convergence to the exact solution; trial solutions may not converge to it at all*.

After adding and as the trial functions the trial solutions become giving the Lagrange functionalThe Ritz-Lagrange system now has two extra variables , and two extra equations. Solving it we find that matches the exact solution as expected.

#### 3. The Ritz-Lagrange Method in Higher Dimensions

The Ritz-Lagrange method described in Section 2 can not be applied to multidimensional boundary value problems. In this section we will develop a suitable generalization and prove that it works. The main distinction is that the boundary operator no longer maps into a finite-dimensional space. Indeed, in dimensions two and higher the boundary values are not arrays of numbers but functions on the boundary forming an infinite-dimensional space . The induction proof of the key Lemma 1 no longer works and its claim itself is false. It is easy to find complete systems of functions with no (finite) nontrivial linear combinations satisfying the boundary conditions.* If we are committed to using arbitrary complete systems of trial functions we must find a way to form their linear combinations that satisfy essential boundary conditions “approximately.”*

To this end we will use a complete system of linear functionals on the Banach space , that is, elements of the dual space (as with , the reader may assume that is a Hilbert space, in which case ). If for all , then and , so we can think of operators as approximations to and the corresponding spaces as approximations to . Assuming is continuous also will be and we can apply Theorem 2 with in place of for each . This gives us a sequence of approximations converging to an exact minimizer of on . The remaining question is whether we can count on to approximate the overall minimizer of on . Before proceeding let us describe the approximating procedure that our approach suggests.

*Multidimensional Ritz-Lagrange Method*. To minimize a functional subject to essential boundary conditions with select* internal trial functions * and* boundary weight functions * with . A* Ritz-Lagrange trial solution * is obtained by solving the system of equations with unknowns , consisting of * internal equations * and * boundary equations *, where is the Lagrange functional and is the Lagrange multiplier.

A justification of our approach is based on Theorem 6. The reader not interested in justification may skip the rest of this section and look at applications in the next one. Even in simplest cases we can not expect to converge to in the same generality as in Theorem 2. The root cause is that* the minimizer in ** is approximated by elements outside of *,* which is why we need ** to be well-behaved on the entire *,* not just *, something avoidable if do satisfy the boundary conditions. In particular,* the values ** are potentially smaller than ** because they are obtained by minimizing ** on larger subspaces *. As a consequence, standard properties of convex functionals, which we relied on in Theorem 2, no longer guarantee convergence of to if converges only weakly (in technical terms, convex functionals are weakly lower semicontinuous, but not necessarily weakly upper semicontinuous [16, III.8.5] and [17, 41.2]).

To make our proof work we need to assume a stronger form of convexity of . For Gateaux differentiable functionals convexity is equivalent to monotonicity of their derivatives; that is, for all , [16, II.5.3]. This is a generalization of a familiar fact that convex functions have monotone derivatives. We will need a form of uniform monotonicity for , compare [16, VI.18.6] and [21, 25.3]; namely,where is a continuous monotone increasing function with . The point is that if then and hence by norm. The next Lemma answers in the affirmative the question about convergence of intermediate minimizers under the uniform monotonicity assumption.

Lemma 5. *Let be a convex Gateaux differentiable functional, and let be a bounded linear map. Let be a complete system in and set . If is weakly coercive on some , and is uniformly monotone on it, then has a minimizer on , as well as minimizers on all with , and .*

Uniform monotonicity also allows us to improve on Theorem 2 by replacing weak limits with strong limits leading to our main result.

Theorem 6. *Let be a convex continuous Gateaux differentiable functional, and let be a bounded linear map. Let be a complete system in , and let be a complete system in . Denote by the linear span of , and set , , and . Suppose that is weakly coercive on some and that is uniformly monotone on it. Then for has unique minimizers , on , , respectively, and in the norm of , while .*

In examples it is typical that does not satisfy (8) on the entire space but does satisfy it on subspaces much larger than , such as . Let us discuss the case of quadratic functionals in more detail. By direct calculation , therefore, We need with to satisfy (8); that is, we need to be strictly positive definite. The multidimensional analog of the functional from our first example is , where is a domain with smooth boundary. It follows from the Poincaré-Friedrichs inequality [20, I.6] and [23] that is strictly positive definite on , but it certainly is not on the entire since for any . Nevertheless, it still follows from the calculus of variations that satisfies (8) on any subspace complementary to the subspace of constants; see, for example, [24, VI.1]. Similar considerations apply to other quadratic forms related to the strongly elliptic equations like the biharmonic equation. They are usually strictly positive definite on complements to finite-dimensional subspaces that they annihilate [23, 22.11]. Such conditions are sometimes called Ker-ellipticity [7, 12].

It should be emphasized that* Theorem 6 does not imply that the double sequence ** converges to *; that is, the repeated limit in it can not be replaced by the double limit. In fact, suppose and let the functionals , where is the dual of , be linearly independent. Then the boundary equations alone are enough to force no matter how large and are. In practice, this means that* one should always take many more internal trial functions than the boundary ones*; hence the prescription . This way for large the approximation will be close to , while in turn will be close to if itself is large enough. The readers familiar with non-conforming Finite Element methods will recognize this as a reflection of the Ladyzhenskaya-Babuška-Brezzi type condition. One of its consequences is that the mesh size on the boundary has to be larger than in the interior, yielding a smaller number of the boundary elements [7, 11]. Modern approach can be found in [25] for finite elements, and in [14] for Galerkin approximations.

As in one-dimensional examples one will have to verify completeness of trial functions in the appropriate space; the same applies to the boundary weights as well. One has to be extra careful with functionals involving higher-order derivatives because the values of function and their derivatives have to be approximated simultaneously. Natural spaces to use are , the spaces of functions with integrable th powers along with all of their derivatives up to order . A generalization of the Weierstrass theorem implies that polynomials form a complete system in for any , any bounded domain , and any positive integer (in fact, polynomials are even uniformly complete [24, II.4.3]). However, polynomials may not always be convenient in a particular problem. The following lemma can be useful in finding other complete systems.

Lemma 7. *Let and be complete systems in and , respectively, where and are some bounded domains. Then the system is complete in .*

If one starts from one-dimensional systems the lemma will only produce complete systems in box-like domains . However, any system of functions complete on a domain will be complete on any of its subdomains, so for an arbitrary domain one can always use a system complete on the smallest box that contains it. A more targeted choice is to take eigenfunctions of an operator on the same domain that is simpler than the one involved but is somewhat similar to it. Various spectral theorems often ensure completeness of eigenfunctions in suitable Sobolev spaces [23, 22.11a].

#### 4. Multidimensional Examples

In this section we illustrate the multidimensional Ritz-Lagrange method developed in Section 3 by applying it to some typical problems. Since calculations by hand quickly become intractable we performed them using a computer algebra system.

Consider a boundary value problem for the Laplace equation in , where is the unit disk, with the boundary condition on and . This equation describes the transverse deflection of a membrane fixed everywhere at the boundary and subjected to pressure given by [24, IV.10.3]. The profile of was chosen so that the problem has an analytic solution which is not a polynomial. Specifically, one can represent the exact solution as a rapidly convergent serieswhere , is the Euler-Mascheroni constant, and is the cosine integral.

To solve the problem we use the multidimensional Ritz-Lagrange method. A variational formulation is to minimize the functional , which gives the total potential energy of the membrane, subject to the boundary condition. In the notation of Section 3 we take with being the restriction of to the boundary . Moreover, is continuous if we take . Our internal trial functions are the monomials, which obviously do not satisfy the boundary condition, and the trial solution is . Note that of Section 3 will be here because of double indexing. As the boundary weight functions we choose the piecewise linear ones on uniform partitions of . Unlike some circle specific choices, for example, the trigonometric functions, such weights can be used on a wide variety of boundaries. Instead of using a single indexed system it is convenient to split it into the constant terms and the linear terms . If the boundary is partitioned into segments, we haveTherefore, the number of boundary weight functions, denoted by in Section 3, will be here. The Lagrange multiplier has the form , and the Lagrange functional is . The unknown coefficients , , and are determined from the system of internal and boundary equations.

The relative errors of the Ritz-Lagrange solutions versus the exact solution (10) are shown in Table 1 as the percentages of the maximum deflection at and . They are quite small considering that one has to determine coefficients in each case. Note that we always keep as recommended in the description of the method to ensure that the system matrices have full rank and are invertible.