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The interior penalty methods using $C^0$ Lagrange elements ($C^0$IPG)developed in the last decade for the fourth order problems are an interestingtopic in academia at present. In this paper, we discuss the adaptive fashion of$C^0$IPG method for the Helmholtz transmission eigenvalue problem.We give the aposteriori error indicators for primal and dual eigenfunctions, and prove theirreliability and efficiency. We also give the a posteriori error indicator foreigenvalues and design a $C^0$IPG adaptive algorithm. Numerical experimentsshow that this algorithm is efficient and can get the optimal convergence rate.

An adaptive C0IPG method for the Helmholtz transmission eigenvalue problem