Sergey Repin

# Abstract

In 2016, the biennial conference Computational Methods in Applied Mathematics (CMAM) was dedicated to a remarkable event: the hundredth anniversary of the Galerkin method. This special volume of the same name journal is mainly based on the papers of participants of this conference. The introductory article contains a brief description of the origin and development of the Galerkin method and gives an overview of the conference, which was held at the University of Jyväskylä (Finland), July 31 – August 6, 2016.

## 1 Galerkin Method

The mathematical idea inherent in the Galerkin method turned out to be extremely fruitful for creating the foundations of both qualitative and quantitative analysis of partial differential equations (PDEs), which began to be intensively studied since the middle of the 19th century. They proved to be much more difficult than ordinary differential equations that were studied earlier. Mathematical models based on PDEs arose from the problems of mechanics and physics. Of course, the question

How to find an approximate solution?

was one of the first.

Already in the late 19th century, it became clear that approximations of the problem u=f (associated with a differential operator acting from a Banach space X to a Banach space Y) should be sought in the form

(1)u(x)uN(x)=i=1Nαiwi(x),

where N is a natural number and the coordinate functions wiXNX form a linearly independent system. It is clear that the weights αi should be chosen from the condition that the residual u-f is minimal (in some sense). At that time, the equations were studied in the framework of classical analysis and, therefore, the space X was considered as the space of continuous functions having sufficient amount of classical derivatives. It is not surprising that one of the first methods that appeared was the collocation method based on the exact satisfaction of the equation at some selected points.

However, in this way it was difficult to create a unified and mathematically justified approach, which required answers to the principal questions:

How to choose the coordinate and test functions and what form of the residual should be used?

Without having the right answers to these questions, it is impossible to prove that uN converges to the exact solution u.

Apparently the very first step in the right direction was made by Walter Ritz (1878–1909) [30] who suggested to define αi by minimization of the variational (energy) functional. The next step was taken by Boris Galerkin (1871–1945) who proposed a much more general approach suitable for partial differential equations of all types. Briefly, the essence of his approach can be formulated as follows:

The approximation is determined from the condition that the residual is orthogonal to a system of linearly independent test functions and the orthogonality relation is understood in the integral sense.

The first mathematical formulation of this method appeared in the paper [14] devoted to a biharmonic type equation (originally, it was published in Russian at the very end of 1915).

### Figure 1

Boris Galerkin and the page in [14] (English translation) containing the main integral relation.

Boris Galerkin was born in Plotsk (nowadays this town is a part of Belarus). Later he moved to St. Petersburg and graduated from St. Petersburg Technological Institute. In 1909, he began to work in higher educational institutions of St. Petersburg (later Leningrad) among which we first of all mention Peter the Great Polytechnic Institute. Yet, as extraordinary as mathematical significance of Galerkin’s ideas, he mainly devoted his creative capacities to mechanics. The invention of the method was motivated by the analysis of mechanical constructions. Galerkin used an earlier publication of Ivan Bubnov (marine officer and engineer), where the method was applied to ship building problems.

In modern notation, the idea of the Galerkin method consists of finding αi by the orthogonality relation

(2)uN-f,ηk=0for all ηkYM,

where uN is defined by (1), the set YMY contains linearly independent test functions in the space conjugate to Y, and , denotes the duality pairing of Y and Y. In general, the sets YM and XN are different (but of course they must be selected such that the system (2) is solvable).

This approach is very flexible. It includes the absolute majority of known methods. If X and Y are Hilbert spaces and the finite dimensional spaces XN and YM coincide, then the method is called Bubnov–Galerkin. This name was used by S. Mikhlin [26] who devoted a lot of time to studying the method and was the first to prove its convergence. If (2) originates from the Euler equation generated by a quadratic functional, then we arrive at the Ritz–Galerkin method. More general schemes with different spaces for coordinate and test functions were studied by G. Petrov who also extended the method to eigenvalue problems. The least squares method and the Fourier method can be also considered as particular forms of (2). The reader can find a systematic consideration of the Galerkin method in application to elliptic, parabolic, and hyperbolic equations in [35].

### Figure 2

W. Ritz, I. Bubnov, and G. Petrov.

## 2 Galerkin Method and Correctness of Boundary Value Problems

The development of the theory of differential equations confirmed the correctness of the path outlined in the works of Galerkin and his successors, which rendered breakthrough results in qualitative and quantitative analysis of PDEs.

At the end of the 19th century it was unclear how to guarantee existence of solutions to boundary value problems for partial differential equations. This question was open even for the simplest boundary value problems. In particular, solvability of the problem

(3)Δu+f=0in Ω,u=0on Ω

was an open problem intensively discussed in the literature (where it was typically understood in the classical sense, i.e., u was considered in C2(Ω)C(Ω¯)). This question was answered after many years of studies that reconstructed the theory of partial differential equations. A new conception of generalized or weak solutions was created by D. Hilbert, H. Poincaré, S. Sobolev, R. Courant, O. A. Ladyzhenskaya, and many other outstanding mathematicians.

In fact, the Galerkin method served as a turning point in the study of this problem. This is easy to see with the paradigm of (3) where the orthogonality relation (2) yields

(4)ΩuNwidx=Ωfwi𝑑xfor all wiXN

and XN contains functions vanishing on the boundary Ω. Let us tend N to +. If we await that uN will tend (in some sense) to the exact solution, then it is natural to consider the limit form of (4), where XN is replaced by the whole space X. This way leads to the generalized statement of (2): find uX satisfying the boundary conditions with square summable u such that

(5)Ωuwdx=Ωfw𝑑xfor all wX,

where X is a proper closure of {XN}.

Integral type definitions of solutions to PDEs appeared in works of many mathematicians. A complete and systematic consideration of this approach is presented in [23, 24, 31]. It is worth noticing that integral type definitions of weak solutions are rather diverse. For example, in 1926, N. Wiener [34] suggested that “admissible” solutions of a linear equation u=f should be considered among the functions satisfying

Ωu*w𝑑x=Ωfw𝑑x

for any sufficiently smooth function w with a compact support in Ω, where * denotes the conjugate operator. This form of the integral identity is close to integral relations used in the so-called weak Galerkin method.

It is easy to show that a sequence of Galerkin solutions converges to a function which satisfies (4) provided that the subspaces {Xk} are limit dense in a reflexive Banach space X (i.e., for any wX, there exists a sequence wkXk such that wkw in X). Indeed, from (4) it follows that uNΩCfΩ with a constant independent of N. Hence, there exists a subsequence of uN weakly converging to a function uX. In view of the limit density property, for any wX there exists a sequence wkX that converges to w in V. Passing to the limit in (4), we conclude that u satisfies (5). We see that Galerkin approximations suggest a simple and efficient method for proving existence of weak solutions to various boundary value problems. E. Hopf used this idea in his famous paper [18] in order to prove existence of weak solutions to nonstationary Navier–Stokes equations. A similar approach was often used by O. Ladyzhenskaya [23] for various nonlinear problems in the theory of viscous fluids and other partial differential equations [24].

It should be noted that roots of the Galerkin method and weak solutions can be found in much earlier studies related to the Virtual Work Principle in mechanics that dates back to J. D’Alembert (who used it in order to derive equations of motion for a mechanical system of rigid bodies) and J.-L. Lagrange (who suggested a more general principle of virtual displacements for continuum media problems). Of course the development of mathematics at that time was insufficient to correctly determine what should be considered as “the set of virtual displacements”.

## 3 Further Development of the Method

Modern numerical methods are based on the development of principle ideas encompassed in the Galerkin method. The amount of related publications is vast, and in this short note it is impossible to present a systematic overview of them. Therefore, we will briefly mention only a few areas that seem to be the most important.

In 1943, R. Courant [12] suggested a version of the method based on locally supported test functions. This was an important step, which generated geometrically flexible numerical schemes with dispersed resolving matrixes. Soon this version of the method gained wide popularity and was named the Finite Element Method (FEM). Mathematical analysis of FEM is based upon two fundamental relations: Galerkin orthogonality and projection estimate or Cea lemma (see, e.g., [7, 10]), which states that the distance between u and the Galerkin solution uN is controlled by the distance between u and the respective finite dimensional space XN.

The projection inequality forms the basis of a priori analysis of the accuracy of Galerkin approximations. The development of this direction was closely connected with the solution of purely mathematical problems related to functional analysis, theory of interpolation, and regularity of weak solutions. It was established that ultimate accuracy of finite-dimensional approximations is determined by the quantity called Kolmogorov’s N width of a compact KX (see [21]):

(6)dN(K)=infXNsupvKinfvNXNv-vNX.

This quantity is extremely important in modern approximation theory, which investigates (6) for various functional and approximation spaces.

The Galerkin method has stimulated other important mathematical studies, especially in numerical linear algebra. It led to the creation of domain decomposition and multigrid methods. In the last decade, methods based on tensor type representations have been of exceptional interest.

In modern computational technologies, we everywhere find traces of the basic Galerkin idea. In mixed and hybrid finite element methods (see, e.g., [8]), it is applied to minimax formulations tested by pairs of functions in the respective primal–dual functional spaces. The discontinuous Galerkin method (DG) [2, 3, 5, 13] uses discontinuous test functions. Many commonly used methods (spectral methods, the finite volume method [15, 25], the weak Galerkin method, isogeometric analysis [11]) can be viewed as advanced versions of the Galerkin method.

The last but not the least direction that is inextricably linked with the Galerkin method is adaptivity. Here, the question

How to reform a subspace XN1 and get a better subspace XN2?

is in the focus of consideration. It is closely related to the theory of mesh adaptation (see, e.g., [29]) and a posteriori error indicators, which are often based on special properties of Galerkin approximations (see, e.g., [4, 9, 27, 32]).

## 4 International Conference CMAM

The above discussed directions and other close topics in applied mathematics and numerical analysis were widely represented at the conference Computational Methods in Applied Mathematics (CMAM-7), which was held in Jyväskylä (Finland) between July 31 and August 6, 2016, and dedicated to the jubilee of the Galerkin method. The conference was organized by the Department of Mathematical Information Technology of the University of Jyväskylä, together with the St. Petersburg Department of the V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences (co-chairs Prof. P. Neittaanmäki and Prof. S. Repin). The conference was supported by the editorial board of the journal Computational Methods in Applied Mathematics and the Radon Institute of Computational and Applied Mathematics of the Austrian Academy of Sciences.

Traditionally, the CMAM conferences focus on various aspects of mathematical modelling and numerical methods for the approximate solution of problems arising in science and engineering and are supported by our journal having the same name. Previous CMAM conferences took place in Minsk (2003, 2007), Trakai (2005), Bedlewo (2010), Berlin (2012), and Strobl (2014).

### Figure 3

Participants of CMAM-7.

Invited keynote lecturers of CMAM-7 were delivered by outstanding scientists in applied mathematics and numerical analysis, namely:

1. Carsten Carstensen (Humboldt University, Berlin, Germany);

2. Roland Glowinski (University of Houston, USA);

3. Boris Khoromskij (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany);

4. Yuri Kuznetsov (Institute of Numerical Mathematics, Moscow, Russia; University of Houston, USA);

5. Raytcho Lazarov (Texas A&M University, USA);

6. Markus Melenk (Technische Universität Wien, Austria);

7. Roderick Melnik (Wilfrid Laurier University, Ontario, Canada);

8. Stefan Sauter (University of Zurich, Switzerland);

9. Rolf Stenberg (Aalto University, Finland);

10. Ragnar Winther (University of Oslo, Norway);

11. Jun Zou (The Chinese University of Hong Kong, Hong Kong).

The conference also included seven minisimposia and many excellent contributed talks. All the information on the conference can be found on www.mit.jyu.fi/scoma/cmam2016/.

The majority of papers of this volume present carefully selected contributions of CMAM participants. The papers by Aghili–Di Pietro–Ruffini [1], Bærland–Lee–Mardal–Winther [6], Guo–Pan–He–Glowinski [16], Gustafsson–Stenberg–Videman [17], and Khoromskij–Repin [20] are related to the development of finite element methods and advanced computational technologies based on ideas of the Galerkin method. In the papers by Gustafsson–Stenberg–Videman [17], Khoromskij–Repin [20], Kruse–Wu [22], and Weymuth–Sauter–Repin [33], the reader will find new results related to error analysis. The paper by Picard [28] is devoted to mathematical foundations of numerical methods for piesoelectric evolutionary models. Tensor type methods for the Hartree–Fock equation are studied in the paper by Khoromskaia–Khoromskij [19]. Kruse–Wu [22] present new approaches to analysis of problems with randomized coefficients.

### References

[1] J. Aghili, D. A. Di Pietro and B. Ruffini, An hp-Hybrid High-Order Method for Variable Diffusion on General Meshes, Comput. Methods Appl. Math. 17 (2017), no. 3, 359–376. 10.1515/cmam-2017-0009Search in Google Scholar

[2] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), 742–760. 10.1137/0719052Search in Google Scholar

[3] D. N. Arnold, F. Brezzi, B. Cockburn and L. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002), no. 5, 1749–1779. 10.1137/S0036142901384162Search in Google Scholar

[4] I. Babuška and W. C. Rheinboldt, A-posteriori error estimates for the finite element method, Internat. J. Numer. Methods Engrg. 12 (1978), 1597–1615. 10.1002/nme.1620121010Search in Google Scholar

[5] I. Babuška and M. Zlamal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal. 10 (1973), 863–875. 10.1137/0710071Search in Google Scholar

[6] T. Bærland, J. J. Lee, K.-A. Mardal and R. Winther, Weakly Imposed Symmetry and Robust Preconditioners for Biot’s Consolidation Model, Comput. Methods Appl. Math. 17 (2017), no. 3, 377–396. 10.1515/cmam-2017-0016Search in Google Scholar

[7] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 1994. 10.1007/978-1-4757-4338-8Search in Google Scholar

[8] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Ser. Comput. Math. 15, Springer, New York, 1991. 10.1007/978-1-4612-3172-1Search in Google Scholar

[9] C. Carstensen and S. Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I: Low order conforming, nonconforming, and mixed FEM, Math. Comp. 71 (2002), no. 239, 945–969. 10.1090/S0025-5718-02-01402-3Search in Google Scholar

[10] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, New York, 1978. 10.1115/1.3424474Search in Google Scholar

[11] J. A. Cottrell, T. Hughes, J. R. Thomas and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, New York, 2009. 10.1002/9780470749081Search in Google Scholar

[12] R. Courant, Variational methods for some problems of equilibrium and vibration, Bull. Amer. Math. Soc. 49 (1943), 1–23. 10.1090/S0002-9904-1943-07818-4Search in Google Scholar

[13] D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Math. Appl. (Berlin) 69, Springer, Berlin, 2011. 10.1007/978-3-642-22980-0Search in Google Scholar

[14] B. G. Galerkin, Beams and plates. Series in some questions of elastic equilibrium of beams and plates (in Russian), Vestnik Ingenerov 19 (1915), 897–908. Search in Google Scholar

[15] S. Godunov, A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations, Math. Sbornik 47 (1959), 271–306. Search in Google Scholar

[16] A. Guo, T.-W. Pan, J. He and R. Glowinski, Numerical Methods for Simulating the Motion of Porous Balls in Simple 3D Shear Flows Under Creeping Conditions, Comput. Methods Appl. Math. 17 (2017), no. 3, 397–412. 10.1515/cmam-2017-0012Search in Google Scholar

[17] T. Gustafsson, R. Stenberg and J. Videman, On Finite Element Formulations for the Obstacle Problem – Mixed and Stabilised Methods, Comput. Methods Appl. Math. 17 (2017), no. 3, 413–429. 10.1515/cmam-2017-0011Search in Google Scholar

[18] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamishen Grundgleichungen, Math. Nachr. 4 (1950/51), 213–231. 10.1002/mana.3210040121Search in Google Scholar

[19] V. Khoromskaia and B. N. Khoromskij, Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach, Comput. Methods Appl. Math. 17 (2017), no. 3, 431–455. 10.1515/cmam-2017-0004Search in Google Scholar

[20] B. Khoromskij and S. Repin, Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems, Comput. Methods Appl. Math. 17 (2017), no. 3, 457–477. 10.1515/cmam-2017-0014Search in Google Scholar

[21] A. Kolmogorov, Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse, Ann. of Math. (2) 37 (1936), no. 1, 107–110. 10.2307/1968691Search in Google Scholar

[22] R. Kruse and Y. Wu, Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients, Comput. Methods Appl. Math. 17 (2017), no. 3, 479–498. 10.1515/cmam-2016-0048Search in Google Scholar

[23] O. A. Ladyzhenskaya, Solution “in the large” of the nonstationary boundary value problem for the Navier–Stokes system with two space variables, Comm. Pure Appl. Math. 12 (1959), 427–433. 10.1002/cpa.3160120303Search in Google Scholar

[24] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Search in Google Scholar

[25] R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2002. 10.1017/CBO9780511791253Search in Google Scholar

[26] S. G. Mikhlin, Variational Methods in Mathematical Physics, Pergamon, Oxford, 1964. Search in Google Scholar

[27] L. A. Oganesjan and L. A. Ruchovec, Study of the rate of convergence of variational difference scheme for second order elliptic equations in two-dimensional region with a smooth boundary, U.S.S.R. Comput. Math. Math. Phys. 9 (1969), 158–183. 10.1016/0041-5553(69)90159-1Search in Google Scholar

[28] R. Picard, On Well-Posedness for a Piezo-Electromagnetic Coupling Model with Boundary Dynamics, Comput. Methods Appl. Math. 17 (2017), no. 3, 499–513. 10.1515/cmam-2017-0005Search in Google Scholar

[29] W. C. Rheinboldt, On a theory of mesh-refinement processes, SIAM J. Numer. Anal. 17 (1980), 766–778. 10.1137/0717064Search in Google Scholar

[30] W. Ritz, Über eine neue Methode zur Lösung gewisser Variationsprobleme der Mathematischen Physik, J. Reine Angew. Math. 135 (1909), 1–61. 10.1515/crll.1909.135.1Search in Google Scholar

[31] S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (in Russian), Izdt. Leningrad. Gos. Univ, Leningrad, 1955; translation in Transl. Math. Monogr. 90, American Mathematical Society, Providence, Rhode Island, 1991. Search in Google Scholar

[32] L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Math. 1605, Springer, Berlin, 1995. 10.1007/BFb0096835Search in Google Scholar

[33] M. Weymuth, S. Sauter and S. Repin, A Posteriori Modelling-Discretization Error Estimate for Elliptic Problems with L-Coefficients, Comput. Methods Appl. Math. 17 (2017), no. 3, 515–531. 10.1515/cmam-2017-0015Search in Google Scholar

[34] N. Wiener, The operational calculus, Math. Ann. 95 (1926), 557–584. 10.1007/BF01206627Search in Google Scholar

[35] E. Zeidler, Nonlinear Functional Analysis and its Applications. II/A: Linear Monotone Operators, Springer, New York, 1990. 10.1007/978-1-4612-0981-2Search in Google Scholar

Published Online: 2017-7-1
Published in Print: 2017-7-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston