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In this paper, we shall study the stability of the Bresse system where the equations are damped by the dissipation from the viscoelasticity and the thermoelasticity. The thermoviscoelastic Bresse beam is a generalization of the thermoviscoelastic Timoshenko beam. Theoretically, we prove the exponential decay of the energy. Later, we introduce and study an implicit Euler type scheme based on finite differences in time and finite elements in space. We show that the discrete energy decreases in time and obtain error estimates. At the end, numerical simulations are presented. K E Y W O R D SBresse system, stability, finite element approximation

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