Context. The Galactic bulge is an intriguing and significant part of our Galaxy, but it is hard to observe because it is both distant and covered by dust in the disk. Therefore, there are not many high-resolution optical spectra of bulge stars with large wavelength coverage, whose determined abundances can be compared with nearby, similarly analyzed stellar samples. Aims: We aim to determine the diagnostically important alpha elements of a sample of bulge giants using high-resolution optical spectra with large wavelength coverage. The abundances found are compared to similarly derived abundances from similar spectra of similar stars in the local thin and thick disks. In this first paper we focus on the solar neighborhood reference sample. Methods: We used spectral synthesis to derive the stellar parameters as well as the elemental abundances of both the local and bulge samples of giants. We took special care to benchmark our method of determining stellar parameters against independent measurements of effective temperatures from angular diameter measurements and surface gravities from asteroseismology. Results: In this first paper we present the method used to determine the stellar parameters and elemental abundances, evaluate them, and present the results for our local disk sample of 291 giants. Conclusions: When comparing our determined spectroscopic temperatures to those derived from angular diameter measurements, we reproduce these with a systematic difference of +10 K and a standard deviation of 53 K. The spectroscopic gravities reproduce those determined from asteroseismology with a systematic offset of +0.10 dex and a standard deviation of 0.12 dex. When it comes to the abundance trends, our sample of local disk giants closely follows trends found in other works analyzing solar neighborhood dwarfs, showing that the much brighter giant stars are as good abundance probes as the often used dwarfs. Based on observations made with the Nordic Optical Telescope (programs 51-018 and 53-002), operated by the Nordic Optical Telescope Scientific Association at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias, and on spectral data retrieved from PolarBase at Observatoire Midi Pyrénées.Full Tables A.1 and A.3 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/598/A100