Variational Analysis and Galerkin Method in Infinite Dimension: Theory, Convergence and Extension for the Poisson
Keywords:
Poisson equation, variational formulation, Galerkin method, finite element method, a priori error estimates, numerical experiments, nonlinear elliptic PDEsAbstract
This paper presents a concise and conceptually unified exposition of the variational formulation and Galerkin finite element discretization of the Poisson equation with Dirichlet boundary conditions. The problem is posed in the Sobolev space , where well-posedness is established via the Lax–Milgram theorem. Standard a priori error estimates are derived, including quasi-optimal convergence in the energy norm through Céa’s lemma and an -error bound obtained by the Aubin–Nitsche duality argument under suitable elliptic regularity assumptions. Reproducible numerical experiments in one dimension using elements and in two dimensions using elements on uniform meshes confirm the optimal convergence rates and . The paper weaves together functional-analytic foundations, finite element assembly, and visual diagnostics into a coherent end-to-end narrative, and briefly explores extensions to semilinear and quasilinear elliptic problems.
References
[1] Gordon D Smith. Numerical solution of partial differential equations: Finite difference methods. Oxford University Press, 1985.
[2] Alfio Quarteroni and Alberto Valli. Numerical approximation of partial differential equations. Springer, 1994.
[3] Philippe G Ciarlet. The finite element method for elliptic problems. NorthHolland, 1978.
[4] Susanne C Brenner and L Ridgway Scott. The mathematical theory of finite element methods. Springer, 2008.
[5] Dietrich Braess. Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge University Press, 2007.
[6] Gilbert Strang and George J Fix. An analysis of the finite element method. Prentice-Hall, 1973.
[7] Claes Johnson. Numerical solution of partial differential equations by the finite element method. Cambridge University Press, 1987.
[8] Thomas J R Hughes. The finite element method: Linear static and dynamic finite element analysis. Dover Publications, 2000.
[9] Junuthula Narasimha Reddy. An introduction to the finite element method. McGraw-Hill, 2005.
[10] Olgierd Cecil Zienkiewicz and Robert Leroy Taylor. The finite element method: The basis. Butterworth-Heinemann, 2000.
[11] Peter D Lax and Arthur N Milgram. Parabolic equations. In Contributions to the theory of partial differential equations, pages 167–190. Princeton University Press, 1954.
[12] Lawrence C Evans. Partial differential equations. American Mathematical Society, 2010
[13] Jean Céa. Approximation variationnelle des problèmes aux limites. Annales de l’Institut Fourier, 14(2):345–444, 1964.
[14] Ivo Babuška. Error-bounds for finite element method. Numerische Mathematik, 16(4):322–333, 1971.
[15] Alexandre Ern and Jean-Luc Guermond. Theory and practice of finite elements. Springer, 2004.
[16] David Gilbarg and Neil S Trudinger. Elliptic partial differential equations of second order. Springer, 2001.
[17] Jacques-Louis Lions and Enrico Magenes. Non-homogeneous boundary value problems and applications. Springer, 1972.
[18] Jean Pierre Aubin. Approximation of elliptic boundary-value problems. Wiley-Interscience, 1972.
[19] Ulrich Trottenberg, Cornelius Oosterlee, and Antonius Schüller. Multigrid. Academic Press, 2001.
[20] Eberhard Zeidler. Applied functional analysis: Applications to mathematical physics. Springer, 1995.
[21] Henri Poincaré. Sur les équations aux dérivées partielles de la physique mathématique. American Journal of Mathematics, 12(3):211–294, 1890.
[22] Robert A Adams. Sobolev spaces. Academic Press, 1975.
[23] Joachim Nitsche. Über ein variationsprinzip zur lösung von dirichletproblemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg, 36:9–15, 1971.
[24] Owe Axelsson and Vaughan A Barker. Finite element solution of boundary value problems: Theory and computation. SIAM, 2001.
[25] Vidar Thomée. Galerkin finite element methods for parabolic problems. Springer, 2006.
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