The Maxwell decoder has been proposed for bridging the gap between the achievable capacity by belief propagation decoding and the maximum a posteriori decoder in binary erasure channels of LDPC codes. The Maxwell decoder, once the belief-propagation decoder gets stuck in a nonempty stopping set, guesses a bit and replicates any running copies of the decoding process. Density evolution and EXIT chart analyses of this iterative decoder show that MAP performance can be derived from the performance of the BP decoder. The complexity of the Maxwell decoder depends exponentially on the number of guesses and a priori we cannot bound the number of guesses, which limits its applicability as a LDPC decoder. In this paper, we adapt the expectation propagation algorithm for LDPC decoding. Our algorithm can be understood as a Maxwell decoder with a bounded complexity. For unbounded complexity it achieves maximum a posteriori decoding. In this paper, we analyze in detail the simplest version of the algorithm, whose complexity is identical to belief propagation, and we demonstrate that the achieved capacity is higher than that of the belief propagation decoder.