Abstract
Most real-world problems depend on time and in this chapter we shall construct numerical methods for solving time dependent differential equations. We do this by first discretizing in space using finite elements, and then in time using finite differences. Various time stepping methods are presented. As model problems we use two classical equations from mathematical physics, namely, the Heat equation, and the Wave equation. Illustrative numerical examples for both equations are presented. To assert the accuracy of the computed solutions we derive both stability estimates, and a priori error estimates. We also formulate space-time finite elements and use them to derive duality based posteriori error estimates.
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W. Bangerth and R. Rannacher. Adaptive Finite Element Methods for Differential Equations. Birkhäuser, 2003.
E. Hairer, S.P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations: Stiff and differential-algebraic problems. Springer series in computational mathematics. Springer-Verlag, 1993.
M. Heath. Scientific Computing. McGraw Hill, 1996.
P. Knabner and L. Angermann. Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer-Verlag, 2000.
J. Strikwerda. Finite Difference Schemes and Partial Differential Equations. SIAM, 2006.
V. Thomée and S. Larsson. Partial Differential Equations with Numerical Methods. Texts in applies Mathematics. Springer, 2005.
A. Valli and A. Quarteroni. Numerical Approximation of Partial Differential Equations. Springer-Verlag, 1994.
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Larson, M.G., Bengzon, F. (2013). Time-Dependent Problems. In: The Finite Element Method: Theory, Implementation, and Applications. Texts in Computational Science and Engineering, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33287-6_5
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DOI: https://doi.org/10.1007/978-3-642-33287-6_5
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Publisher Name: Springer, Berlin, Heidelberg
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Online ISBN: 978-3-642-33287-6
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