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As  ε  goes to zero, the unique solution of the scalar advection‐diffusion equationy    t    ε    −  ε    y    x  x    ε    +  M    y    x    ε    =  0  , ( x , t )∈(0,1)×(0, T ) with Dirichlet boundary conditions exhibits a boundary layer of size   O  (  ε  )   and an internal layer of size   O  (    ε    )  . If the time  T  is large enough, these thin layers, where the solution  y   ε   displays rapid variations, intersect and interact with each other. Using the method of matched asymptotic expansions, we show how we can construct an explicit approximationP   ˜εof the solution  y   ε   satisfying   ‖    y    ε    −P   ˜ε‖L    ∞    (  0  ,  T  ;    L    2    (  0  ,  1  )  )    =  O  (    ε    3  /  2    )   and   ‖    y    ε    −P   ˜ε‖L    2    (  0  ,  T  ;    H    1    (  0  ,  1  )  )    =  O  (  ε  )   for all  ε  small enough.

Asymptotic analysis of an advection‐diffusion equation involving interacting boundary and internal layers