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Euler's backward formula; regularity; Rothe's method; coupled consolidation of clay; coupled dynamical thermoelasticity; convergence; semidiscrete approximate solutions; time discretization; discretization in space; finite element method; weakest assumptions
The authors study problems of existence and uniqueness of solutions of various variational formulations of the coupled problem of dynamical thermoelasticity and of the convergence of approximate solutions of these problems. First, the semidiscrete approximate solutions is defined, which is obtained by time discretization of the original variational problem by Euler's backward formula. Under certain smoothness assumptions on the date authors prove existence and uniqueness of the solution and establish the rate of convergence $O(\Delta t^{1/2})$ of Rothe's functions in the spaces $C(I;W\frac12(\Omega))$ and $C(I;L_2(\Omega))$ for the displacement components and the temperature, respectively. Regularity of solutions is discussed. In Part 2 the authors define the fully discretized solution of the original variational problem by Euler's backward formula and the simplest finite elements. Convergence of these approximate solutions is proved. In Part 3, the weakest assumptions possible are imposed onto the data, which corresponds to a different definition of the variational solution. Existence and uniqueness of the variational solution, as well as convergence of the fully discretized solutions, are proved.
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